Solving Polynomials

What does it mean to solve an equation?

Methods of Solving….

Solving Polynomials

• Factoring Options1. More Difficult GCF Factoring

2. Sum or Difference of Cubes

3. Factor by Grouping

4. U Substitution

5. Polynomial Division (to factor out a binomial term)

GCF Factoring

1)

2)

24 483 xx

086 23 xx

086 23 xx

0432 2 xx

086 23 xx

0432 2 xx

02 2 x 043 x

086 23 xx

0432 2 xx

02 2 x 043 x

0x4

3x

Sum or Difference of Cubes

1. Solve

2. Solve

083 x

0273 x

0273 x

0933 2 xxx

0273 x

0933 2 xxx

03 x 0932 xx

0273 x

0933 2 xxx

03 x 0932 xx

3x 2

333

2

273

2

91493 ix

Factor by Grouping

3)

4)

0933 23 xxx

0632 23 xxx

0632 23 xxx

0632 23 xxx

0632 23 xxx

0632 23 xxx

02322 xxx

0632 23 xxx

0632 23 xxx

02322 xxx

0232 xx

0632 23 xxx

0632 23 xxx

02322 xxx

0232 xx

2 3

032

xx

x

Solving an Equation of Quadratic Type (“U” Substitution)

5)

6)

023 24 xx

0365 24 xx

0365 24 xx

049 22 xx

0365 24 xx

049 22 xx

092 x 042 x

0365 24 xx

049 22 xx

092 x 042 x

92 x 42 x

0365 24 xx

049 22 xx

092 x 042 x

92 x 42 x

ix 3 2x

Practice

• Polynomials WS

• Return Tests from Last Week

Synthetic Division• Use synthetic division to find the quotient

and the remainder when is divided by x – 2. If x – 2 is a factor, then factor the polynomial completely.

• How can we determine whether x – 2 is a factor?

3 26 19 16 4x x x

Factor Theorem

• A polynomial f(x) has a factor x – a iff the remainder is 0.

Example 1

• Use synthetic division to determine whether x – 1 is a factor of x³ - 1.

Example 2• x = -4 is a solution of x³ - 28x – 48 = 0.

Use synthetic division to factor and find all remaining solutions.

Example 3• x + 3 is a factor of y = 3x³ + 2x² - 19x + 6.

Find all the zeros of this polynomial.

Rational Roots (Zeros) Test

• Every rational zero that is possible for a given polynomial can be expressed as the factors of the constant term divided by the factors of the leading coefficient.

Example 4• List all possible rational roots for the

polynomial y = 10x³ - 15x² - 16x + 12. Then, divide out the factor and solve for all remaining zeros.

Example 5• List all possible rational roots for the

polynomial y = x³ - 7x – 6. Then, divide out the factor and solve for all remaining zeros.

Practice

• Pg. 213 (53 – 67 odd, 68)

• Pg. 278 (41, 43, 55, 57, 59)