Question 1

Find all the rational zeros of the function $f ( x ) = x ^ { 5 } - 5 x ^ { 4 } + 11 x ^ { 3 } - 23 x ^ { 2 } + 28 x - 12$ .
A)​x = -2,2,1,3
B)​x = 3,1
C)​x = -2,-1,-3
D)​x = -1,-2
E)x = -2,2,-1,1,-3

Accepted Answer

The answer of Find all the rational zeros of the...

Question 2

Find all rational roots of the equation. 147x ³ - 98x ² + 27x - 18 = 0 A) $x = \frac { 2 } { 5 }$ B) $x = \frac { 2 } { 3 }$ C) $x = - \frac { 3 } { 7 } i$ D) $x = \frac { 3 } { 7 } i$

Accepted Answer

The answer of Find all rational roots of the equation....

Question 3

The number (1 + i)is a root of the equation. X ³ - 10x ² + 18x - 16 = 0 Find the other roots of the equation. A)x = 1 - i B)x = 1 + i C)x = 5 D)x = 8

Accepted Answer

The answer of The number (1 + i)is a root...

Question 4

​Use synthetic division to divide. $$\frac { 4 x ^ { 3 } + 31 x ^ { 2 } + 52 x - 15 } { x - \frac { 1 } { 4 } }$$

Accepted Answer

The answer of ​Use synthetic division to divide. \[\frac {...

Question 5

Use synthetic division to perform the division.​ $\frac { 3 x ^ { 3 } + 4 x ^ { 2 } + 18 x - 25 } { x - 1 }$

Accepted Answer

The answer of Use synthetic division to perform the division.​...

Question 6

Use synthetic division to perform the division.​ $\frac { 3 x ^ { 4 } - 7 x ^ { 3 } - 16 x + 5 } { x - 3 }$ ​
A) $x ^ { 3 } + x ^ { 2 } + 3 x + 2 - \frac { 11 } { x - 3 }$
B) $3 x ^ { 3 } + 2 x ^ { 2 } + 6 x + 2 + \frac { 11 } { x - 3 }$
C) $3 x ^ { 3 } - 2 x ^ { 2 } + 6 x - 2 + \frac { 11 } { x - 3 }$
D) $3 x ^ { 3 } + 2 x ^ { 2 } + 6 x + 2 + 11$
E)none of these

Accepted Answer

The answer of Use synthetic division to perform the division.​...

Question 7

Find all real zeros of the polynomial $f ( x ) = x ^ { 4 } - 113 x ^ { 2 } + 3136$ and determine the multiplicity of each.
A)x = 49,multiplicity 2;x = 64,multiplicity 2
B)x = 7,multiplicity 2;x = 8,multiplicity 2
C)x = 49,multiplicity 2;x = 8,multiplicity 1
D)x = -7,multiplicity 2;x = -8,multiplicity 2
E)x = 7,multiplicity 1;x = -7,multiplicity 1;x = 8,multiplicity 1;x = -8,multiplicity 1

Accepted Answer

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

Question 8

Use the zero or root feature of a graphing utility to approximate the zeros of $f ( x ) = 2 x ^ { 3 } + 8 x ^ { 2 } - 5 x - 7$ accurate to the nearest thousandth.
A)​x = -4.692
B)​x = -8.492,-0.695,1.186
C)​x = -3.575
D)​x = -4.388,-0.720,1.108
E)​x = -3.910,-0.763,1.173

Accepted Answer

The zeros of the polynomial can be found using a graphing utility's zero or root feature, which will show where the function crosses the x-axis. The correct zeros, rounded to the nearest thousandth, are -4.388, -0.720, and 1.108, as these values are where the function $f(x) = 2x^3 + 8x^2 - 5x - 7$ intersects the x-axis.

Question 9

Use Descartes' rule of signs to find the number of possible positive,negative,and nonreal roots for the following equation. 4x ⁷ + 9x ² + 5x + 10 = 0 A)0 positive;1 negative;6 nonreal B)2 positive;1 or 3 negative;4 or 0 nonreal C)0 positive;0 negative;7 nonreal D)0 positive;0 or 2 negative;5 or 7 nonreal E)none of these

Accepted Answer

Descartes' rule of signs for the given polynomial $4x^{17} + 9x^{12} + 5x + 10 = 0$ indicates 0 changes in sign for positive roots, suggesting 0 positive roots. For negative roots, substituting $x = -x$ gives $4(-x)^{17} + 9(-x)^{12} + 5(-x) + 10$, which results in 2 sign changes, indicating 2 or 0 negative roots (since the number of negative roots is either the number of sign changes or less than that by an even number). The total number of roots for a 17th degree polynomial is 17, so if there are 0 positive and 2 or 0 negative roots, there must be 17 or 15 nonreal roots, respectively. None of the provided options accurately reflects this analysis.

Question 10

Let $P ( x ) = 2 x ^ { 3 } + 9 x ^ { 2 } - 7 x + 3$ .
​
Use synthetic division to find the value P (-7).

Accepted Answer

The answer of Let \(P ( x ) = 2...

Question 11

Find all rational roots of the equation. 13x ³ - 2x ² + 52x - 8 = 0 A) $x = \frac { 2 } { 3 }$ B)x = -2i C)x = 2i D) $x = - \frac { 2 } { 13 }$ E) $x = \frac { 2 } { 13 }$

Accepted Answer

The answer of Find all rational roots of the equation....

Question 12

Find all the real zeros of $f ( x ) = 2 x ^ { 3 } - 6 x ^ { 2 } + 2 x - 6$ .
A)​x = 3, $- \frac { 1 } { 3 }$
B)​x = 2,1,-6
C)​x = ± 1,3
D)​x = 3
E)​x = $\frac { 1 } { 2 }$ , $- \frac { 1 } { 6 }$

Accepted Answer

The polynomial can be factored or solved using the Rational Root Theorem, and the only real zero is x = 3.

Question 13

Find all the rational zeros of the function $f ( x ) = - 2 x ^ { 4 } - x ^ { 3 } + 51 x ^ { 2 } + 25 x - 25$ .
A)x = 1,-5,5,-2
B)x = -2,-5,-5
C)x = -1,-5,5, $\frac { 1 } { 2 }$
D)​x = $- \frac { 1 } { 5 }$ , $\frac { 5 } { 2 }$ , $\frac { 1 } { 2 }$ ,5
E)​x = -2,-5, $\frac { 5 } { 2 }$ , $\frac { 1 } { 2 }$

Accepted Answer

The answer of Find all the rational zeros of the...

Question 14

Use Descartes' Rule of Signs to determine the possible number of positive and negative zeros of $f ( x ) = 4 x ^ { 3 } + 3 x ^ { 2 } + 5 x + 1$ .
A)3 positive reals or 1 positive real;3 negative reals or 1 negative real
B)3 positive reals or 1 positive real;no negative reals
C)0 positive real;3 negative reals or 1 negative real
D)1 positive real;3 negative reals or 1 negative real
E)no positive reals;no negative reals

Accepted Answer

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

Question 15

Use Descartes' Rule of Signs to determine the possible number of positive and negative zeros of $f ( x ) = 2 x ^ { 3 } - 2 x ^ { 2 } + 3 x - 4$ .
A)3 positive reals or 1 positive real;3 negative reals or 1 negative real
B)3 positive reals or 1 positive real;no negative reals
C)1 positive real;3 negative reals or 1 negative real
D)3 positive reals or 1 positive real;1 negative real
E)no positive reals;no negative reals

Accepted Answer

Descartes' Rule of Signs for positive zeros: Change signs between terms in $f(x)$. There's 1 change (from $+3x$ to $-4$), so 1 positive real zero. For negative zeros, substitute $x$ with $-x$ to get $f(-x) = -2x^3-2x^2-3x-4$, which has no sign changes, indicating no negative real zeros.

Question 16

Find all the real zeros of $f ( x ) = 4 x ^ { 3 } - 20 x ^ { 2 } + 17 x - 4$ .
A)x = $\frac { 1 } { 2 }$ ,4
B)x = $- \frac { 1 } { 2 }$ , $- \frac { 1 } { 4 }$
C)x = $\frac { 1 } { 4 }$ ,-2
D)x = $\frac { 1 } { 2 }$ , $- \frac { 1 } { 4 }$
E)x = $\frac { 1 } { 2 }$ ,-4

Accepted Answer

To find the real zeros of the polynomial $f(x) = 4x^3 - 20x^2 + 17x - 4$, we can use the Rational Root Theorem, synthetic division, or factoring if possible. Factoring by grouping or trying possible rational roots (factors of the constant term divided by factors of the leading coefficient) can lead to finding that $x = \frac{1}{2}$ and $x = 4$ are zeros of the polynomial. Substituting these values into the polynomial confirms they are indeed roots.

Quiz 13: Polynomial and Synthetic Division

Find all the rational zeros of the function $f ( x ) = x ^ { 5 } - 5 x ^ { 4 } + 11 x ^ { 3 } - 23 x ^ { 2 } + 28 x - 12$ .

Find all rational roots of the equation. 147x ³ - 98x ² + 27x - 18 = 0

The number (1 + i) is a root of the equation. X ³ - 10x ² + 18x - 16 = 0 Find the other roots of the equation.

Use synthetic division to divide. $\frac { 4 x ^ { 3 } + 31 x ^ { 2 } + 52 x - 15 } { x - \frac { 1 } { 4 } }$

Use synthetic division to perform the division. $\frac { 3 x ^ { 3 } + 4 x ^ { 2 } + 18 x - 25 } { x - 1 }$

Use synthetic division to perform the division. $\frac { 3 x ^ { 4 } - 7 x ^ { 3 } - 16 x + 5 } { x - 3 }$

Find all real zeros of the polynomial $f ( x ) = x ^ { 4 } - 113 x ^ { 2 } + 3136$ and determine the multiplicity of each.

Use the zero or root feature of a graphing utility to approximate the zeros of $f ( x ) = 2 x ^ { 3 } + 8 x ^ { 2 } - 5 x - 7$ accurate to the nearest thousandth.

Use Descartes' rule of signs to find the number of possible positive,negative,and nonreal roots for the following equation. 4x ⁷ + 9x ² + 5x + 10 = 0

Let $P ( x ) = 2 x ^ { 3 } + 9 x ^ { 2 } - 7 x + 3$ . Use synthetic division to find the value P (-7).

Find all rational roots of the equation. 13x ³ - 2x ² + 52x - 8 = 0

Find all the real zeros of $f ( x ) = 2 x ^ { 3 } - 6 x ^ { 2 } + 2 x - 6$ .

Find all the rational zeros of the function $f ( x ) = - 2 x ^ { 4 } - x ^ { 3 } + 51 x ^ { 2 } + 25 x - 25$ .

Use Descartes' Rule of Signs to determine the possible number of positive and negative zeros of $f ( x ) = 4 x ^ { 3 } + 3 x ^ { 2 } + 5 x + 1$ .

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

Find all the real zeros of $f ( x ) = 4 x ^ { 3 } - 20 x ^ { 2 } + 17 x - 4$ .

Quiz 13: Polynomial and Synthetic Division

Find all the rational zeros of the function f(x) =x5−5x4+11x3−23x2+28x−12f ( x ) = x ^ { 5 } - 5 x ^ { 4 } + 11 x ^ { 3 } - 23 x ^ { 2 } + 28 x - 12f(x) =x5−5x4+11x3−23x2+28x−12 .

Find all rational roots of the equation. ​ 147x 3 - 98x 2 + 27x - 18 = 0 ​

The number (1 + i) is a root of the equation. ​ X 3 - 10x 2 + 18x - 16 = 0 ​ Find the other roots of the equation. ​

​Use synthetic division to divide. 4x3+31x2+52x−15x−14\frac { 4 x ^ { 3 } + 31 x ^ { 2 } + 52 x - 15 } { x - \frac { 1 } { 4 } }x−41​4x3+31x2+52x−15​

Use synthetic division to perform the division.​ 3x3+4x2+18x−25x−1\frac { 3 x ^ { 3 } + 4 x ^ { 2 } + 18 x - 25 } { x - 1 }x−13x3+4x2+18x−25​

Use synthetic division to perform the division.​ 3x4−7x3−16x+5x−3\frac { 3 x ^ { 4 } - 7 x ^ { 3 } - 16 x + 5 } { x - 3 }x−33x4−7x3−16x+5​ ​

Find all real zeros of the polynomial f(x) =x4−113x2+3136f ( x ) = x ^ { 4 } - 113 x ^ { 2 } + 3136f(x) =x4−113x2+3136 and determine the multiplicity of each.

Use the zero or root feature of a graphing utility to approximate the zeros of f(x) =2x3+8x2−5x−7f ( x ) = 2 x ^ { 3 } + 8 x ^ { 2 } - 5 x - 7f(x) =2x3+8x2−5x−7 accurate to the nearest thousandth.

Use Descartes' rule of signs to find the number of possible positive,negative,and nonreal roots for the following equation. 4x 7 + 9x 2 + 5x + 10 = 0

Let P(x)=2x3+9x2−7x+3P ( x ) = 2 x ^ { 3 } + 9 x ^ { 2 } - 7 x + 3P(x)=2x3+9x2−7x+3 . ​ Use synthetic division to find the value P (-7).

Find all rational roots of the equation. ​ 13x 3 - 2x 2 + 52x - 8 = 0 ​

Find all the real zeros of f(x) =2x3−6x2+2x−6f ( x ) = 2 x ^ { 3 } - 6 x ^ { 2 } + 2 x - 6f(x) =2x3−6x2+2x−6 .

Find all the rational zeros of the function f(x) =−2x4−x3+51x2+25x−25f ( x ) = - 2 x ^ { 4 } - x ^ { 3 } + 51 x ^ { 2 } + 25 x - 25f(x) =−2x4−x3+51x2+25x−25 .

Use Descartes' Rule of Signs to determine the possible number of positive and negative zeros of f(x) =4x3+3x2+5x+1f ( x ) = 4 x ^ { 3 } + 3 x ^ { 2 } + 5 x + 1f(x) =4x3+3x2+5x+1 .

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

Find all the real zeros of f(x) =4x3−20x2+17x−4f ( x ) = 4 x ^ { 3 } - 20 x ^ { 2 } + 17 x - 4f(x) =4x3−20x2+17x−4 .