Download Polynomial Terms and Operations

Polynomial Terms and
Operations
EXAMPLE 1
Add polynomials vertically and horizontally
a. Add 2x3 – 5x2 + 3x – 9 and x3 + 6x2 + 11 in a vertical format.
SOLUTION
a.
2x3 – 5x2 + 3x – 9
+ x3 + 6x2
+ 11
3x3 + x2 + 3x + 2
EXAMPLE 1
Add polynomials vertically and horizontally
b. Add 3y3 – 2y2 – 7y and –4y2 + 2y – 5 in a horizontal
format.
(3y3 – 2y2 – 7y) + (–4y2 + 2y – 5)
= 3y3 – 2y2 – 4y2 – 7y + 2y – 5
= 3y3 – 6y2 – 5y – 5
EXAMPLE 2
Subtract polynomials vertically and horizontally
a. Subtract 3x3 + 2x2 – x + 7 from 8x3 – x2 – 5x + 1 in a vertical format.
SOLUTION
a. Align like terms, then add the opposite of the
subtracted polynomial.
8x3 – x2 – 5x + 1
– (3x3 + 2x2 – x + 7)
8x3 – x2 – 5x + 1
+ – 3x3 – 2x2 + x – 7
5x3 – 3x2 – 4x – 6
EXAMPLE 2
Subtract polynomials vertically and horizontally
b. Subtract 5z2 – z + 3 from 4z2 + 9z – 12 in a horizontal
format.
Write the opposite of the subtracted polynomial,
then add like terms.
(4z2 + 9z – 12) – (5z2 – z + 3) = 4z2 + 9z – 12 – 5z2 + z – 3
= 4z2 – 5z2 + 9z + z – 12 – 3
= –z2 + 10z – 15
for Examples 1 and 2
GUIDED PRACTICE
Find the sum or difference.
1. (t2 – 6t + 2) + (5t2 – t – 8)
ANSWER
6t2 – 7t – 6
2. (8d – 3 + 9d3) – (d3 – 13d2 – 4)
ANSWER
8d3 + 13d2 + 8d + 1
EXAMPLE 3
Multiply polynomials vertically and horizontally
a. Multiply –2y2 + 3y – 6 and y – 2 in a vertical format.
b. Multiply x + 3 and 3x2 – 2x + 4 in a horizontal format.
SOLUTION
a.
–2y2+ 3y – 6
y –2
4y2 – 6y + 12
Multiply –2y2 + 3y – 6 by –2 .
–2y3 + 3y2 – 6y
Multiply –2y2 + 3y – 6 by y
–2y3 + 7y2 –12y + 12
Combine like terms.
EXAMPLE 3
Multiply polynomials vertically and horizontally
b. (x + 3)(3x2 – 2x + 4) = (x + 3)3x2 – (x + 3)2x + (x + 3)4
= 3x3 + 9x2 – 2x2 – 6x + 4x + 12
= 3x3 + 7x2 – 2x + 12
EXAMPLE 4
Multiply three binomials
Multiply x – 5, x + 1, and x + 3 in a horizontal format.
(x – 5)(x + 1)(x + 3) = (x2 – 4x – 5)(x + 3)
= (x2 – 4x – 5)x + (x2 – 4x – 5)3
= x3 – 4x2 – 5x + 3x2 – 12x – 15
= x3 – x2 – 17x – 15
EXAMPLE 5
Use special product patterns
a. (3t + 4)(3t – 4)
= (3t)2 – 42
Sum and difference
= 9t2 – 16
b. (8x – 3)2
= (8x)2 – 2(8x)(3) + 32
Square of a binomial
= 64x2 – 48x + 9
c. (pq + 5)3
= (pq)3 + 3(pq)2(5) + 3(pq)(5)2 + 53
= p3q3 + 15p2q2 + 75pq + 125
Cube of a
binomial
for Examples 3, 4 and 5
GUIDED PRACTICE
Find the product.
3. (x + 2)(3x2 – x – 5)
ANSWER
3x3 + 5x2 – 7x – 10
4. (a – 5)(a + 2)(a + 6)
ANSWER
(a3 + 3a2 – 28a – 60)
5. (xy – 4)3
ANSWER
x3y3 – 12x2y2 + 48xy – 64