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Math  /  Algebra

QuestionUse synthetic division to divide. SEE EXAMPLE 2
19. x4−25x2+144x^{4}-25 x^{2}+144 divided by x−4x-4
20. x3+6x2+3x−10x^{3}+6 x^{2}+3 x-10 divided by x+5x+5
21. x5+2x4−3x3+x−1x^{5}+2 x^{4}-3 x^{3}+x-1 divided by x+2x+2
22. −x4+7x3+x2−2x−12-x^{4}+7 x^{3}+x^{2}-2 x-12 divided by x−3x-3
23. Use synthetic division to show that the remainder of f(x)=x4−6x3−33x2+46x+75f(x)=x^{4}-6 x^{3}-33 x^{2}+46 x+75 divided by x−9x-9 is P(9)P(9). SEE EXAMPLE 3

Studdy Solution

STEP 1

1. We are using synthetic division to divide polynomials.
2. Synthetic division is applicable when dividing by a linear polynomial of the form x−c x - c .
3. The remainder theorem states that the remainder of the division of a polynomial f(x) f(x) by x−c x - c is f(c) f(c) .

STEP 2

1. Perform synthetic division for each polynomial division problem.
2. Verify the remainder using the remainder theorem for problem 23.

STEP 3

For problem 19, divide x4−25x2+144 x^4 - 25x^2 + 144 by x−4 x - 4 using synthetic division.
- Write the coefficients of the polynomial: [1,0,−25,0,144] [1, 0, -25, 0, 144] . - The divisor is x−4 x - 4 , so c=4 c = 4 . - Set up the synthetic division:
410−250144416−36−14414−9−360\begin{array}{c|ccccc} 4 & 1 & 0 & -25 & 0 & 144 \\ & & 4 & 16 & -36 & -144 \\ \hline & 1 & 4 & -9 & -36 & 0 \\ \end{array}
- The quotient is x3+4x2−9x−36 x^3 + 4x^2 - 9x - 36 and the remainder is 0 0 .

STEP 4

For problem 20, divide x3+6x2+3x−10 x^3 + 6x^2 + 3x - 10 by x+5 x + 5 .
- Write the coefficients: [1,6,3,−10] [1, 6, 3, -10] . - The divisor is x+5 x + 5 , so c=−5 c = -5 . - Set up the synthetic division:
−5163−10−5−51011−20\begin{array}{c|cccc} -5 & 1 & 6 & 3 & -10 \\ & & -5 & -5 & 10 \\ \hline & 1 & 1 & -2 & 0 \\ \end{array}
- The quotient is x2+x−2 x^2 + x - 2 and the remainder is 0 0 .

STEP 5

For problem 21, divide x5+2x4−3x3+x−1 x^5 + 2x^4 - 3x^3 + x - 1 by x+2 x + 2 .
- Write the coefficients: [1,2,−3,0,1,−1] [1, 2, -3, 0, 1, -1] . - The divisor is x+2 x + 2 , so c=−2 c = -2 . - Set up the synthetic division:
−212−301−1−206−122210−36−1121\begin{array}{c|cccccc} -2 & 1 & 2 & -3 & 0 & 1 & -1 \\ & & -2 & 0 & 6 & -12 & 22 \\ \hline & 1 & 0 & -3 & 6 & -11 & 21 \\ \end{array}
- The quotient is x4−3x2+6x−11 x^4 - 3x^2 + 6x - 11 and the remainder is 21 21 .

STEP 6

For problem 22, divide −x4+7x3+x2−2x−12 -x^4 + 7x^3 + x^2 - 2x - 12 by x−3 x - 3 .
- Write the coefficients: [−1,7,1,−2,−12] [-1, 7, 1, -2, -12] . - The divisor is x−3 x - 3 , so c=3 c = 3 . - Set up the synthetic division:
3−171−2−12−31239111−14133799\begin{array}{c|ccccc} 3 & -1 & 7 & 1 & -2 & -12 \\ & & -3 & 12 & 39 & 111 \\ \hline & -1 & 4 & 13 & 37 & 99 \\ \end{array}
- The quotient is −x3+4x2+13x+37 -x^3 + 4x^2 + 13x + 37 and the remainder is 99 99 .

STEP 7

For problem 23, verify the remainder of f(x)=x4−6x3−33x2+46x+75 f(x) = x^4 - 6x^3 - 33x^2 + 46x + 75 divided by x−9 x - 9 is f(9) f(9) .
- Write the coefficients: [1,−6,−33,46,75] [1, -6, -33, 46, 75] . - The divisor is x−9 x - 9 , so c=9 c = 9 . - Set up the synthetic division:
91−6−334675927−54−7213−6−83\begin{array}{c|ccccc} 9 & 1 & -6 & -33 & 46 & 75 \\ & & 9 & 27 & -54 & -72 \\ \hline & 1 & 3 & -6 & -8 & 3 \\ \end{array}
- The remainder is 3 3 , which confirms that f(9)=3 f(9) = 3 .

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