Prentice Hall Algebra 2

Chapter 5 Polynomials and Polynomial Functions > 5-4 Dividing Polynomials

If the width of the box is 15 in., what are the other two dimensions?

The width of the box is x plus 5 . So if x plus 5 equals 15 comma then x equals 10 .

Substitute for x to find the length and height (or depth).

Length: x + 8 = 10 + 8 = 18 in . cap lengthcolon , x plus 8 equals 10 plus 8 equals 18 , in .

Height: x − 6 = 10 − 6 = 4 in . cap heightcolon , x minus 6 equals 10 minus 6 equals 4 in .

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If the polynomial x 3 + 6 x 2 + 11 x + 6 x cubed , plus 6 , x squared , plus 11 x plus 6 expresses the volume, in cubic inches, of the box, and the width is open x plus 1 close in., what are the dimensions of the box?

The Remainder Theorem provides a quick way to find the remainder of a polynomial long-division problem.

Take Note Theorem: The Remainder Theorem

If you divide a polynomial P(x) of degree n greater than or equal to 1 by x minus eh comma then the remainder is P(a).

Here's Why It Works When you divide polynomial P(x) by D(x), you find P ( x ) = D ( x ) Q ( x ) + R ( x ) . p open x close equals d open x close q open x close plus r open x close .

P ( x ) = ( x − a ) Q ( x ) + R ( x ) Substitute ( x − a ) for D ( x ) . P ( a ) = ( a − a ) Q ( a ) + R ( a ) Evaluate P ( a ) . Substitute a for x . = R ( a ) Simplify . table with 3 rows and 3 columns , row1 column 1 , p , open x close , column 2 equals . open , x minus eh , close q , open x close , plus r , open x close , column 3 cap substitute . open , x minus eh , close . for d open x close . , row2 column 1 , p , open eh close , column 2 equals . open , eh minus eh , close q , open eh close , plus r , open eh close , column 3 cap evaluate p open eh close . . cap substitute . eh , for , x . , row3 column 1 , , column 2 equals r , open eh close , column 3 cap simplify , . , end table

Problem 5 Evaluating a Polynomial

GRIDDED RESPONSE

Given that P ( x ) = x 5 − 2 x 3 − x 2 + 2 , p open x close equals , x to the fifth , minus 2 , x cubed , minus , x squared , plus 2 comma what is p open 3 close question mark

By the Remainder Theorem, P(3) is the remainder when you divide P(x) by x minus 3 .

3 1 0 − 2 − 1 0 2 3 9 21 60 180 1 3 7 20 60 182 ¯ table with 3 rows and 1 column , row1 column 1 , 3 1 0 minus 2 minus 1 0 2 , row2 column 1 , 3 9 21 60 , 180 , row3 column 1 , 1 3 7 20 60 , 182 bar , end table

p open 3 close equals 182 .

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Given that P ( x ) = x 5 − 3 x 4 − 28 x 3 + 5 x + 20 , p open x close equals , x to the fifth , minus , 3 x to the fourth , minus , 28 x cubed , plus 5 x plus 20 comma what is p open negative 4 close question mark

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Take Note Theorem: The Remainder Theorem

Problem 5 Evaluating a Polynomial

Think

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