Question 1

Find a polynomial of least degree with only real coefficients and having the given zeros.
-$3 , - 11$, and $4 + 4 i$

Accepted Answer

Given the zeros $3, -11$, and $4 + 4i$, the polynomial must also have the conjugate of $4 + 4i$, which is $4 - 4i$, to ensure all coefficients are real. The polynomial with these zeros is obtained by multiplying $(x - 3)(x + 11)(x - (4 + 4i))(x - (4 - 4i))$. Simplifying this product leads to a polynomial of degree 4 with real coefficients, matching option A.

Question 2

Find a polynomial of least degree with only real coefficients and having the given zeros.
-$1 - \sqrt { 3 } , 1 + \sqrt { 3 }$, and $1 + i$

Accepted Answer

The answer of Find a polynomial of least degree with...

Question 3

Find a polynomial of least degree with only real coefficients and having the given zeros.
-$2 + i , 4$

Accepted Answer

The answer of Find a polynomial of least degree with...

Question 4

Find a polynomial of least degree with only real coefficients and having the given zeros.
-$- \sqrt { 3 } , \sqrt { 3 }$, and 8 (multiplicity 2 )

Accepted Answer

Given zeros are $-\sqrt{3}, \sqrt{3}$, and $8$ with multiplicity 2. The polynomial with these zeros can be constructed as $(x + \sqrt{3})(x - \sqrt{3})(x - 8)^2$. Simplifying, $(x^2 - 3)(x - 8)^2$ leads to a polynomial of degree 4, which matches the form of option A after full expansion and combining like terms.

Question 5

Find all complex zeros of the polynomial function. Give exact values. List multiple zeros as necessary.
-$$f ( x ) = x ^ { 4 } - 10 x ^ { 3 } + 23 x ^ { 2 } - 10 x + 22$$

Accepted Answer

The answer of Find all complex zeros of the polynomial...

Question 6

Use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of negative real zeros for the function.
-$$f ( x ) = - 2 x ^ { 4 } - 4 x ^ { 3 } - 4 x ^ { 2 } - 3 x + 8$$

Accepted Answer

The answer of Use Descartes' Rule of Signs to determine...

Question 7

Use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of negative real zeros for the function.
-$$f ( x ) = 8 x ^ { 5 } - 7 x ^ { 4 } + 8 x ^ { 3 } - 6$$

Accepted Answer

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Question 8

Find a polynomial of least degree with only real coefficients and having the given zeros.
-$2 - i , - 6$

Accepted Answer

The polynomial must include the conjugate of the complex zero, $2 + i$, to have real coefficients. Thus, the zeros are $2 - i$, $2 + i$, and $-6$. The polynomial is $(x - (2-i))(x - (2+i))(x + 6)$, which simplifies to $x^3 + 2x^2 - 19x + 30$.

Question 9

Use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of negative real zeros for the function.
-
f ( x ) = 9 x ^ { 6 } - 8 x ^ { 4 } - 3 x ^ { 3 } + 9 x ^ { 2 } - 8 x

Accepted Answer

The answer of Use Descartes' Rule of Signs to determine...

Question 10

Use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of negative real zeros for the function.
-
f ( x ) = - 2 x ^ { 4 } + 9 x ^ { 3 } - 7 x ^ { 2 } + 9 x - 7

Accepted Answer

The answer of Use Descartes' Rule of Signs to determine...

Question 11

Use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of negative real zeros for the function.
-$$f ( x ) = 2 x ^ { 8 } + 4 x ^ { 6 } + 2 x ^ { 4 } + 7 x ^ { 2 } + 2$$

Accepted Answer

The answer of Use Descartes' Rule of Signs to determine...

Question 12

Find a polynomial of degree 3 with real coefficients that satisfies the given conditions.
Zero of  -2  having multiplicity 2 and zero of 4 having multiplicity  1 ; P(0)=-16

Accepted Answer

The polynomial can be expressed as $P(x) = a(x + 2)^2(x - 4)$. Given $P(0) = -16$, solve for $a$: $-16 = a(0 + 2)^2(0 - 4)$, which gives $a = 1$. Thus, $P(x) = (x + 2)^2(x - 4) = x^3 - 12x - 16$.

Question 13

Find all complex zeros of the polynomial function. Give exact values. List multiple zeros as necessary.
-$$f ( x ) = x ^ { 3 } - 7 x ^ { 2 } + 15 x - 25$$

Accepted Answer

To find the complex zeros of the given polynomial $f(x) = x^3 - 7x^2 + 15x - 25$, we can start by attempting to factor it or use the Rational Root Theorem to find a real root. By inspection or trial and error, we can see that $x = 5$ is a root because $f(5) = 5^3 - 7(5)^2 + 15(5) - 25 = 125 - 175 + 75 - 25 = 0$. After finding $x = 5$ as a root, we can perform polynomial division or use synthetic division to factor the polynomial as $(x - 5)(x^2 - 2x + 5)$. The quadratic formula can be used to find the roots of the quadratic factor $x^2 - 2x + 5$, which are $x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(5)}}{2(1)} = \frac{2 \pm \sqrt{4 - 20}}{2} = \frac{2 \pm \sqrt{-16}}{2} = 1 \pm 2i$.Therefore, the complex zeros of the polynomial are $5, 1 + 2i, 1 - 2i$.

Question 14

Use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of negative real zeros for the function.
-$$f ( x ) = 2 x ^ { 3 } - 8 x ^ { 2 } + 8 x + 2$$

Accepted Answer

The answer of Use Descartes' Rule of Signs to determine...

Question 15

Find all complex zeros of the polynomial function. Give exact values. List multiple zeros as necessary.
-$$f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 5 x + 39$$

Accepted Answer

The answer of Find all complex zeros of the polynomial...

Question 16

Find a polynomial of degree 3 with real coefficients that satisfies the given conditions.
Zeros of  -5  and  1 - i, P(0)=10

Accepted Answer

Since the polynomial has real coefficients and a complex zero of $1 - i$, its conjugate $1 + i$ must also be a zero. The zeros are -5, $1 - i$, and $1 + i$. The polynomial can be formed by multiplying $(x + 5)(x - (1 - i))(x - (1 + i))$. Simplifying this product gives a polynomial, and adjusting the constant term to ensure $P(0) = 10$ leads to the correct polynomial $P(x) = x^3 + 3x^2 - 8x + 10$.

Question 17

Find a polynomial of degree 3 with real coefficients that satisfies the given conditions.
Zeros of 5 and i,P(1)=-8

Accepted Answer

Since the polynomial has real coefficients and a complex zero $i$, its conjugate $-i$ must also be a zero. The zeros are 5, $i$, and $-i$. The polynomial can be constructed as $(x-5)(x-i)(x+i)$, which simplifies to $x^3 - 5x^2 + x - 5$. Checking $P(1) = 1 - 5 + 1 - 5 = -8$ confirms option C.

Question 18

Find a polynomial of least degree with only real coefficients and having the given zeros.
-$- \sqrt { 3 } , \sqrt { 3 }$, and $- 4 \mathrm { i }$

Accepted Answer

Given the zeros $-\sqrt{3}, \sqrt{3},$ and $-4i$, we must also include the conjugate of the complex zero, $4i$, to ensure real coefficients. The factors corresponding to these zeros are $x + \sqrt{3}, x - \sqrt{3}, x + 4i,$ and $x - 4i$. Multiplying these factors gives the polynomial $f(x) = (x^2 - 3)(x^2 + 16) = x^4 + 13x^2 - 48$.

Question 19

Find a polynomial of least degree with only real coefficients and having the given zeros.
-$4 + \sqrt { 2 } , 4 - \sqrt { 2 }$, and 3

Accepted Answer

The polynomial with the given zeros $4 + \sqrt{2}$, $4 - \sqrt{2}$, and 3 is found by multiplying the factors $(x - (4 + \sqrt{2}))(x - (4 - \sqrt{2}))(x - 3)$. This simplifies to $(x - 4)^2 - 2$ times $(x - 3)$, which results in the polynomial $x^3 - 11x^2 + 38x - 42$.

Question 20

Find a polynomial of least degree with only real coefficients and having the given zeros.
-$6 + 5 i$ and $6 - 5 i$

Accepted Answer

If a polynomial has real coefficients, then its non-real roots must come in conjugate pairs. Given the roots $6 + 5i$ and $6 - 5i$, we can form the polynomial by multiplying $(x - (6 + 5i))(x - (6 - 5i))$, which simplifies to $x^2 - 12x + 61$.

Quiz 4: Polynomials and Rational Functions

Find a polynomial of least degree with only real coefficients and having the given zeros. - $3 , - 11$ , and $4 + 4 i$

Find a polynomial of least degree with only real coefficients and having the given zeros. - $1 - \sqrt { 3 } , 1 + \sqrt { 3 }$ , and $1 + i$

Find a polynomial of least degree with only real coefficients and having the given zeros. - $2 + i , 4$

Find a polynomial of least degree with only real coefficients and having the given zeros. - $- \sqrt { 3 } , \sqrt { 3 }$ , and 8 (multiplicity 2 )

Find all complex zeros of the polynomial function. Give exact values. List multiple zeros as necessary. - $f ( x ) = x ^ { 4 } - 10 x ^ { 3 } + 23 x ^ { 2 } - 10 x + 22$

Use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of negative real zeros for the function. - $f ( x ) = - 2 x ^ { 4 } - 4 x ^ { 3 } - 4 x ^ { 2 } - 3 x + 8$

Use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of negative real zeros for the function. - $f ( x ) = 8 x ^ { 5 } - 7 x ^ { 4 } + 8 x ^ { 3 } - 6$

Find a polynomial of least degree with only real coefficients and having the given zeros. - $2 - i , - 6$

Use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of negative real zeros for the function. - f ( x ) = 9 x ^ { 6 } - 8 x ^ { 4 } - 3 x ^ { 3 } + 9 x ^ { 2 } - 8 x

Use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of negative real zeros for the function. - f ( x ) = - 2 x ^ { 4 } + 9 x ^ { 3 } - 7 x ^ { 2 } + 9 x - 7

Use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of negative real zeros for the function. - $f ( x ) = 2 x ^ { 8 } + 4 x ^ { 6 } + 2 x ^ { 4 } + 7 x ^ { 2 } + 2$

Find a polynomial of degree 3 with real coefficients that satisfies the given conditions. Zero of -2 having multiplicity 2 and zero of 4 having multiplicity 1 ; P(0) =-16

Find all complex zeros of the polynomial function. Give exact values. List multiple zeros as necessary. - $f ( x ) = x ^ { 3 } - 7 x ^ { 2 } + 15 x - 25$

Use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of negative real zeros for the function. - $f ( x ) = 2 x ^ { 3 } - 8 x ^ { 2 } + 8 x + 2$

Find all complex zeros of the polynomial function. Give exact values. List multiple zeros as necessary. - $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 5 x + 39$

Find a polynomial of degree 3 with real coefficients that satisfies the given conditions. Zeros of -5 and 1 - i, P(0) =10

Find a polynomial of degree 3 with real coefficients that satisfies the given conditions. Zeros of 5 and i,P(1) =-8

Find a polynomial of least degree with only real coefficients and having the given zeros. - $- \sqrt { 3 } , \sqrt { 3 }$ , and $- 4 \mathrm { i }$

Find a polynomial of least degree with only real coefficients and having the given zeros. - $4 + \sqrt { 2 } , 4 - \sqrt { 2 }$ , and 3

Find a polynomial of least degree with only real coefficients and having the given zeros. - $6 + 5 i$ and $6 - 5 i$

Quiz 4: Polynomials and Rational Functions

Find a polynomial of least degree with only real coefficients and having the given zeros. - 3,−113 , - 113,−11 , and 4+4i4 + 4 i4+4i

Find a polynomial of least degree with only real coefficients and having the given zeros. - 1−3,1+31 - \sqrt { 3 } , 1 + \sqrt { 3 }1−3​,1+3​ , and 1+i1 + i1+i

Find a polynomial of least degree with only real coefficients and having the given zeros. - 2+i,42 + i , 42+i,4

Find a polynomial of least degree with only real coefficients and having the given zeros. - −3,3- \sqrt { 3 } , \sqrt { 3 }−3​,3​ , and 8 (multiplicity 2 )

Find all complex zeros of the polynomial function. Give exact values. List multiple zeros as necessary. - f(x) =x4−10x3+23x2−10x+22f ( x ) = x ^ { 4 } - 10 x ^ { 3 } + 23 x ^ { 2 } - 10 x + 22f(x) =x4−10x3+23x2−10x+22

Use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of negative real zeros for the function. - f(x) =−2x4−4x3−4x2−3x+8f ( x ) = - 2 x ^ { 4 } - 4 x ^ { 3 } - 4 x ^ { 2 } - 3 x + 8f(x) =−2x4−4x3−4x2−3x+8

Use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of negative real zeros for the function. - f(x) =8x5−7x4+8x3−6f ( x ) = 8 x ^ { 5 } - 7 x ^ { 4 } + 8 x ^ { 3 } - 6f(x) =8x5−7x4+8x3−6

Find a polynomial of least degree with only real coefficients and having the given zeros. - 2−i,−62 - i , - 62−i,−6

Use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of negative real zeros for the function. - f ( x ) = 9 x ^ { 6 } - 8 x ^ { 4 } - 3 x ^ { 3 } + 9 x ^ { 2 } - 8 x

Use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of negative real zeros for the function. - f ( x ) = - 2 x ^ { 4 } + 9 x ^ { 3 } - 7 x ^ { 2 } + 9 x - 7

Use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of negative real zeros for the function. - f(x) =2x8+4x6+2x4+7x2+2f ( x ) = 2 x ^ { 8 } + 4 x ^ { 6 } + 2 x ^ { 4 } + 7 x ^ { 2 } + 2f(x) =2x8+4x6+2x4+7x2+2

Find a polynomial of degree 3 with real coefficients that satisfies the given conditions. Zero of -2 having multiplicity 2 and zero of 4 having multiplicity 1 ; P(0) =-16

Find all complex zeros of the polynomial function. Give exact values. List multiple zeros as necessary. - f(x) =x3−7x2+15x−25f ( x ) = x ^ { 3 } - 7 x ^ { 2 } + 15 x - 25f(x) =x3−7x2+15x−25

Use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of negative real zeros for the function. - f(x) =2x3−8x2+8x+2f ( x ) = 2 x ^ { 3 } - 8 x ^ { 2 } + 8 x + 2f(x) =2x3−8x2+8x+2

Find all complex zeros of the polynomial function. Give exact values. List multiple zeros as necessary. - f(x) =x3−3x2−5x+39f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 5 x + 39f(x) =x3−3x2−5x+39

Find a polynomial of degree 3 with real coefficients that satisfies the given conditions. Zeros of -5 and 1 - i, P(0) =10

Find a polynomial of degree 3 with real coefficients that satisfies the given conditions. Zeros of 5 and i,P(1) =-8

Find a polynomial of least degree with only real coefficients and having the given zeros. - −3,3- \sqrt { 3 } , \sqrt { 3 }−3​,3​ , and −4i- 4 \mathrm { i }−4i

Find a polynomial of least degree with only real coefficients and having the given zeros. - 4+2,4−24 + \sqrt { 2 } , 4 - \sqrt { 2 }4+2​,4−2​ , and 3

Find a polynomial of least degree with only real coefficients and having the given zeros. - 6+5i6 + 5 i6+5i and 6−5i6 - 5 i6−5i

Find a polynomial of least degree with only real coefficients and having the given zeros. - $3 , - 11$ , and $4 + 4 i$

Find a polynomial of least degree with only real coefficients and having the given zeros. - $1 - \sqrt { 3 } , 1 + \sqrt { 3 }$ , and $1 + i$

Find a polynomial of least degree with only real coefficients and having the given zeros. - $2 + i , 4$

Find a polynomial of least degree with only real coefficients and having the given zeros. - $- \sqrt { 3 } , \sqrt { 3 }$ , and 8 (multiplicity 2 )

Find all complex zeros of the polynomial function. Give exact values. List multiple zeros as necessary. - $f ( x ) = x ^ { 4 } - 10 x ^ { 3 } + 23 x ^ { 2 } - 10 x + 22$

Use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of negative real zeros for the function. - $f ( x ) = - 2 x ^ { 4 } - 4 x ^ { 3 } - 4 x ^ { 2 } - 3 x + 8$

Use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of negative real zeros for the function. - $f ( x ) = 8 x ^ { 5 } - 7 x ^ { 4 } + 8 x ^ { 3 } - 6$

Find a polynomial of least degree with only real coefficients and having the given zeros. - $2 - i , - 6$

Use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of negative real zeros for the function. - $f ( x ) = 2 x ^ { 8 } + 4 x ^ { 6 } + 2 x ^ { 4 } + 7 x ^ { 2 } + 2$

Find all complex zeros of the polynomial function. Give exact values. List multiple zeros as necessary. - $f ( x ) = x ^ { 3 } - 7 x ^ { 2 } + 15 x - 25$

Use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of negative real zeros for the function. - $f ( x ) = 2 x ^ { 3 } - 8 x ^ { 2 } + 8 x + 2$

Find all complex zeros of the polynomial function. Give exact values. List multiple zeros as necessary. - $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 5 x + 39$

Find a polynomial of least degree with only real coefficients and having the given zeros. - $- \sqrt { 3 } , \sqrt { 3 }$ , and $- 4 \mathrm { i }$

Find a polynomial of least degree with only real coefficients and having the given zeros. - $4 + \sqrt { 2 } , 4 - \sqrt { 2 }$ , and 3

Find a polynomial of least degree with only real coefficients and having the given zeros. - $6 + 5 i$ and $6 - 5 i$