BuyFind

Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071
BuyFind

Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

Solutions

Chapter 1.4, Problem 36E
To determine

To calculate: The simplified form of the expression 2x23x2x212x2+5x+2x2+x2

Expert Solution

Answer to Problem 36E

The simplified form of the expression is (x2)(2x+1)_

Explanation of Solution

Given information:

The expressionis 2x23x2x212x2+5x+2x2+x2

Formula used:

If a, b, c and d are any numbers then abcd is expressed by a×bc×d .

The difference of squares of two numbers is algebraically expressed as: A2B2=(A+B)(AB)

To factor a quadratic polynomial of the form x2+bx+c , note that (x+r)(x+s)=x2+(r+s)x+rs So,need to choose numbers r and s So, that r+s=b and rs=c .

To factor a quadratic polynomial of the form ax2+bx+c with a1 look for factors of the form of (px+r) and (qx+s)

  ax2+bx+c=(px+r)(qx+s)=pqx2+(ps+qr)x+rs

Therefore, try to find numbers p,q,r and s such that pq=a,rs=c,ps+qr=c.

Calculation:

Consider the provided expression 2x23x2x212x2+5x+2x2+x2

Recall that if a, b, c and d are any numbers then abcd is expressed by a×bc×d .

The difference of squares of two numbers is algebraically expressed as: A2B2=(A+B)(AB)

To factor a quadratic polynomial of the form x2+bx+c ,note that (x+r)(x+s)=x2+(r+s)x+rs So, need to choose numbers r and s So that r+s=b and rs=c .

To factor a quadratic polynomial of the form ax2+bx+c with a1 look for factors of the form of (px+r) and (qx+s)

  ax2+bx+c=(px+r)(qx+s)=pqx2+(ps+qr)x+rs

Therefore, try to find numbers p,q,r and s such that pq=a,rs=c,ps+qr=c.

  2x23x2x212x2+5x+2x2+x2=2x23x2x21×x2+x22x2+5x+2

Factor the numerator and denominator,

  2x23x2x212x2+5x+2x2+x2=(x2)(x+1)(x+1)(x1)×(x+2)(x1)(x+2)(2x+1)=(x+2)(x+1)(x1)(x+2)(x+1)(x1)(x2)(2x+1)

Cancel out the common factors,

  2x23x2x212x2+5x+2x2+x2=(x2)(2x+1)

Thus, the simplified form of the expression is (x2)(2x+1) .

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