Solved on Dec 17, 2023

Find the quadratic polynomial that completes the factorization of $64 v^{3} + 125 = (4 v + 5)(\square)$ .

STEP 1

Assumptions
1. We are given a factor of a cubic polynomial $64v^3 + 125$ which is $(4v + 5)$ .
2. We need to find the quadratic polynomial that when multiplied by $(4v + 5)$ gives the original cubic polynomial.
3. The quadratic polynomial can be expressed in the form $av^2 + bv + c$ .

STEP 2

To find the quadratic polynomial, we will perform polynomial long division, dividing the cubic polynomial by the given linear factor.

STEP 3

Set up the polynomial long division.
$\begin{array}{r|l} 4v + 5 & 64v^3 + 0v^2 + 0v + 125 \\ \end{array}$

STEP 4

Determine the first term of the quotient by dividing the leading term of the cubic polynomial by the leading term of the linear factor.
$\frac{64v^3}{4v} = 16v^2$

STEP 5

Multiply the entire linear factor $(4v + 5)$ by the term found in STEP_4, $16v^2$ .
$(4v + 5)(16v^2) = 64v^3 + 80v^2$

STEP 6

Subtract the result of the multiplication from the cubic polynomial.
$\begin{array}{r|l} 4v + 5 & 64v^3 + 0v^2 + 0v + 125 \\ \hline & 64v^3 + 80v^2 \\ \end{array}$
$- (64v^3 + 80v^2)$
$0v^3 - 80v^2 + 0v + 125$

STEP 7

Bring down the next term to continue the division.
$-80v^2 + 0v + 125$

STEP 8

Determine the next term of the quotient by dividing the leading term of the new polynomial by the leading term of the linear factor.
$\frac{-80v^2}{4v} = -20v$

STEP 9

Multiply the entire linear factor $(4v + 5)$ by the term found in STEP_8, $-20v$ .
$(4v + 5)(-20v) = -80v^2 - 100v$

STEP 10

Subtract the result of the multiplication from the new polynomial.
$\begin{array}{r|l} 4v + 5 & 64v^3 + 0v^2 + 0v + 125 \\ \hline & 64v^3 + 80v^2 \\ & - (64v^3 + 80v^2) \\ \hline & 0v^3 - 80v^2 + 0v + 125 \\ & -80v^2 - 100v \\ \end{array}$
$- (-80v^2 - 100v)$
$0v^2 + 100v + 125$

STEP 11

Bring down the next term to continue the division.
$100v + 125$

STEP 12

Determine the next term of the quotient by dividing the leading term of the new polynomial by the leading term of the linear factor.
$\frac{100v}{4v} = 25$

STEP 13

Multiply the entire linear factor $(4v + 5)$ by the term found in STEP_12, $25$ .
$(4v + 5)(25) = 100v + 125$

STEP 14

Subtract the result of the multiplication from the new polynomial.
$\begin{array}{r|l} 4v + 5 & 64v^3 + 0v^2 + 0v + 125 \\ \hline & 64v^3 + 80v^2 \\ & - (64v^3 + 80v^2) \\ \hline & 0v^3 - 80v^2 + 0v + 125 \\ & -80v^2 - 100v \\ & - (-80v^2 - 100v) \\ \hline & 0v^2 + 100v + 125 \\ & 100v + 125 \\ & - (100v + 125) \\ \hline & 0v + 0 \\ \end{array}$

STEP 15

The remainder is zero, which means the division is exact. The quotient of the division is the quadratic polynomial we are looking for.
$Quadratic\, polynomial = 16v^2 - 20v + 25$

STEP 16

Therefore, the quadratic polynomial that completes the factorization is $16v^2 - 20v + 25$ .
$64v^3 + 125 = (4v + 5)(16v^2 - 20v + 25)$

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Find the quadratic polynomial that completes the factorization of $64 v^{3} + 125 = (4 v + 5)(\square)$ .

STEP 1

Assumptions
1. We are given a factor of a cubic polynomial $64v^3 + 125$ which is $(4v + 5)$ .
2. We need to find the quadratic polynomial that when multiplied by $(4v + 5)$ gives the original cubic polynomial.
3. The quadratic polynomial can be expressed in the form $av^2 + bv + c$ .

STEP 2

To find the quadratic polynomial, we will perform polynomial long division, dividing the cubic polynomial by the given linear factor.

STEP 3

Set up the polynomial long division.
$\begin{array}{r|l} 4v + 5 & 64v^3 + 0v^2 + 0v + 125 \\ \end{array}$

STEP 4

Determine the first term of the quotient by dividing the leading term of the cubic polynomial by the leading term of the linear factor.
$\frac{64v^3}{4v} = 16v^2$

STEP 5

Multiply the entire linear factor $(4v + 5)$ by the term found in STEP_4, $16v^2$ .
$(4v + 5)(16v^2) = 64v^3 + 80v^2$

STEP 6

Subtract the result of the multiplication from the cubic polynomial.
$\begin{array}{r|l} 4v + 5 & 64v^3 + 0v^2 + 0v + 125 \\ \hline & 64v^3 + 80v^2 \\ \end{array}$
$- (64v^3 + 80v^2)$
$0v^3 - 80v^2 + 0v + 125$

STEP 7

Bring down the next term to continue the division.
$-80v^2 + 0v + 125$

STEP 8

Determine the next term of the quotient by dividing the leading term of the new polynomial by the leading term of the linear factor.
$\frac{-80v^2}{4v} = -20v$

STEP 9

Multiply the entire linear factor $(4v + 5)$ by the term found in STEP_8, $-20v$ .
$(4v + 5)(-20v) = -80v^2 - 100v$

STEP 10

Subtract the result of the multiplication from the new polynomial.
$\begin{array}{r|l} 4v + 5 & 64v^3 + 0v^2 + 0v + 125 \\ \hline & 64v^3 + 80v^2 \\ & - (64v^3 + 80v^2) \\ \hline & 0v^3 - 80v^2 + 0v + 125 \\ & -80v^2 - 100v \\ \end{array}$
$- (-80v^2 - 100v)$
$0v^2 + 100v + 125$

STEP 11

Bring down the next term to continue the division.
$100v + 125$

STEP 12

Determine the next term of the quotient by dividing the leading term of the new polynomial by the leading term of the linear factor.
$\frac{100v}{4v} = 25$

STEP 13

Multiply the entire linear factor $(4v + 5)$ by the term found in STEP_12, $25$ .
$(4v + 5)(25) = 100v + 125$

STEP 14

STEP 15

The remainder is zero, which means the division is exact. The quotient of the division is the quadratic polynomial we are looking for.
$Quadratic\, polynomial = 16v^2 - 20v + 25$

STEP 16

Therefore, the quadratic polynomial that completes the factorization is $16v^2 - 20v + 25$ .
$64v^3 + 125 = (4v + 5)(16v^2 - 20v + 25)$

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Find the quadratic polynomial that completes the factorization of 64v3+125=(4v+5)(□)64 v^{3} + 125 = (4 v + 5)(\square)64v3+125=(4v+5)(□).

STEP 1

STEP 2

To find the quadratic polynomial, we will perform polynomial long division, dividing the cubic polynomial by the given linear factor.

STEP 3

Set up the polynomial long division.4v+564v3+0v2+0v+125 \begin{array}{r|l} 4v + 5 & 64v^3 + 0v^2 + 0v + 125 \\ \end{array} 4v+5​64v3+0v2+0v+125​

STEP 4

Determine the first term of the quotient by dividing the leading term of the cubic polynomial by the leading term of the linear factor.64v34v=16v2 \frac{64v^3}{4v} = 16v^2 4v64v3​=16v2

STEP 5

Multiply the entire linear factor (4v+5)(4v + 5)(4v+5) by the term found in STEP_4, 16v216v^216v2.(4v+5)(16v2)=64v3+80v2 (4v + 5)(16v^2) = 64v^3 + 80v^2 (4v+5)(16v2)=64v3+80v2

STEP 6

STEP 7

Bring down the next term to continue the division.−80v2+0v+125 -80v^2 + 0v + 125 −80v2+0v+125

STEP 8

Determine the next term of the quotient by dividing the leading term of the new polynomial by the leading term of the linear factor.−80v24v=−20v \frac{-80v^2}{4v} = -20v 4v−80v2​=−20v

STEP 9

Multiply the entire linear factor (4v+5)(4v + 5)(4v+5) by the term found in STEP_8, −20v-20v−20v.(4v+5)(−20v)=−80v2−100v (4v + 5)(-20v) = -80v^2 - 100v (4v+5)(−20v)=−80v2−100v

STEP 10

STEP 11

Bring down the next term to continue the division.100v+125 100v + 125 100v+125

STEP 12

Determine the next term of the quotient by dividing the leading term of the new polynomial by the leading term of the linear factor.100v4v=25 \frac{100v}{4v} = 25 4v100v​=25

STEP 13

Multiply the entire linear factor (4v+5)(4v + 5)(4v+5) by the term found in STEP_12, 252525.(4v+5)(25)=100v+125 (4v + 5)(25) = 100v + 125 (4v+5)(25)=100v+125

STEP 14

STEP 15

The remainder is zero, which means the division is exact. The quotient of the division is the quadratic polynomial we are looking for.Quadratic polynomial=16v2−20v+25 Quadratic\, polynomial = 16v^2 - 20v + 25 Quadraticpolynomial=16v2−20v+25

STEP 16

Therefore, the quadratic polynomial that completes the factorization is 16v2−20v+2516v^2 - 20v + 2516v2−20v+25.64v3+125=(4v+5)(16v2−20v+25) 64v^3 + 125 = (4v + 5)(16v^2 - 20v + 25) 64v3+125=(4v+5)(16v2−20v+25)

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Find the quadratic polynomial that completes the factorization of $64 v^{3} + 125 = (4 v + 5)(\square)$ .

Set up the polynomial long division.
$\begin{array}{r|l} 4v + 5 & 64v^3 + 0v^2 + 0v + 125 \\ \end{array}$

Determine the first term of the quotient by dividing the leading term of the cubic polynomial by the leading term of the linear factor.
$\frac{64v^3}{4v} = 16v^2$

Multiply the entire linear factor $(4v + 5)$ by the term found in STEP_4, $16v^2$ .
$(4v + 5)(16v^2) = 64v^3 + 80v^2$

Bring down the next term to continue the division.
$-80v^2 + 0v + 125$

Determine the next term of the quotient by dividing the leading term of the new polynomial by the leading term of the linear factor.
$\frac{-80v^2}{4v} = -20v$

Multiply the entire linear factor $(4v + 5)$ by the term found in STEP_8, $-20v$ .
$(4v + 5)(-20v) = -80v^2 - 100v$

Bring down the next term to continue the division.
$100v + 125$

Determine the next term of the quotient by dividing the leading term of the new polynomial by the leading term of the linear factor.
$\frac{100v}{4v} = 25$

Multiply the entire linear factor $(4v + 5)$ by the term found in STEP_12, $25$ .
$(4v + 5)(25) = 100v + 125$

The remainder is zero, which means the division is exact. The quotient of the division is the quadratic polynomial we are looking for.
$Quadratic\, polynomial = 16v^2 - 20v + 25$

Therefore, the quadratic polynomial that completes the factorization is $16v^2 - 20v + 25$ .
$64v^3 + 125 = (4v + 5)(16v^2 - 20v + 25)$