1. 5.3 WARM-UP Decide whether the function is a polynomial function.
If so, write it in standard form and state its degree, type and leading coefficient.
Not a function
yes
Degree: 5
Leading coefficient: -8
Standard form:
Use direct substitution to evaluate the following polynomials
-49 76
Use synthetic substitution to evaluate each polynomial
-49
149

2. Add, Subtract, and Multiply Polynomials
5.3
Add, Subtract, and Multiply Polynomials
To add or subtract polynomials , combine the coefficients of like terms.
a. Add 2x3 – 5x2 + 3x – 9 and x3 + 6x in a vertical format.
2x3 – 5x2 + 3x – 9
+ x3 + 6x
3x3
+ x2
+ 3x
+ 2
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

3. 8x3 – x2 – 5x + 1 – (3x3 + 2x2 – x + 7) 8x3 – x2 – 5x + 1
c. Subtract 3x3 + 2x2 – x + 7 from 8x3 – x2 – 5x + 1 in a vertical format.
(Align like terms, then change all the signs on the second equation, then add)
8x3 – x2 – 5x + 1
– (3x3 + 2x2 – x + 7)
8x3 – x2 – 5x + 1
+ – 3x3 – 2x2 + x – 7
5x3
– 3x2
– 4x
– 6
d. 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
= – z2
+ 10z
– 15

4. Find the sum or difference.
1. (t2 – 6t + 2) + (5t2 – t – 8)
t2 – 6t + 2
+ 5t2 – t – 8
6t2
– 7t
– 6
2. (8d – 3 + 9d3) – (d3 – 13d2 – 4)
= 8d – 3 + 9d3 – d3 + 13d2 + 4
= 8d3
+ 13d2
+ 8d
+ 1

5. – 2y2 + 3y – 6 y – 2 4y2 – 6y + 12 – 2y3 + 3y2 – 6y – 2y3 +7y2 –12y
Multiply – 2y2 + 3y – 6 and y – 2 in a vertical format.
– 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.
Multiply x + 3 and 3x2 – 2x + 4 in a horizontal format.
(x + 3)(3x2 – 2x + 4)
= 3x3
– 2x2
+ 4x
+ 9x2
– 6x
+ 12
= 3x3
+ 7x2
– 2x
+ 12

6. Multiply three binomials
Multiply x – 5, x + 1, and x + 3 in a horizontal format.
(x – 5)(x + 1)(x + 3) =
(x2
+1x
– 5x
– 5)
(x + 3)
(x2 – 4x – 5)(x + 3) = x3
– 4x2
– 5x
+ 3x2
– 12x
– 15
(x + 3)(x2 – 4x – 5) =
= x3
– x2
– 17x
– 15

7. Try finding the product.
(x + 2)(3x2 – x – 5)
(a – 5)(a + 2)(a + 6)
3x2 – x – 5
x + 2
= (a2 – 3a – 10)(a + 6)
= (a2 – 3a – 10)a + (a2 – 3a – 10)6
6x2 – 2x – 10
3x3 – x2 – 5x
= (a3 – 3a2 – 10a + 6a2 – 18a – 60)
3x3 + 5x2 – 7x – 10
= (a3 + 3a2 – 28a – 60)
= (xy – 4)
(xy – 4)
(xy – 4)3
= (x2y2 – 4xy – 4xy + 16)
(xy – 4)
= (x2y2 – 8xy + 16)
(xy – 4)
= (x3y3 – 8x2y2 + 16xy – 4x2y xy – 64 )
= x3y3 – 12x2y2 + 48xy – 64

8. Use special product patterns
Try these:
a. (3t + 4)(3t – 4)
= (3t)2 – 42
Sum and difference
*You can just square the first number and square the last number
= 9t2 – 16
b. (8x – 3)2
= (8x)2 – 2(8x)(3) + 32
Square of a binomial
= 64x2 – 48x + 9
*square the first number, square the last number and multiply the first and last numbers and double
HOMEWORK 5.3
p. 349 #3-25 all, EOP