Polynomials Worksheet Page 3
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58-4 Solving Polynomial Equations
Quick Review
Exercises
Find the real or imaginary solutions of each equation
One way to solve a polynomial equation is by factoring.
First write the equation in the form P (x) = 0, where P (x)
by factoring.
is the polynomial. Then factor the polynomial. Last, use
30. x
2
- 11x = -24
2
= -4x - 1
31. 4x
the Zero-Product Property to find the solutions, or roots.
3
+ 3x
2
= 27x
2
+ 3 = 4x
32. 3x
33. 2x
The solutions may be real or imaginary. Real solutions and
approximations of irrational solutions can also be found by
Find the real roots of each equation by graphing.
using a graphing calculator.
34. x
4
+ 3x
2
- 2x + 5 = 0
Example
35. x
2
+ 3 = x
3
- 5
Solve x
3
+ 4x
2
= 12x by factoring.
36. The height and width of a rectangular prism are each
x
3
+ 4x
2
- 12x = 0
Subtract 12x from each side.
2 inches shorter than the length of the prism. The
x(x - 2)(x + 6) = 0
volume of the prism is 40 cubic inches. Approximate the
Factor the left side.
dimensions of the prism to the nearest hundredth.
x = 0, x - 2 = 0, x + 6 = 0
Zero-Product Property
x = 0, x = 2, x = - 6
Solve each equation.
The solutions are 0, 2, and -6.
8-5 Dividing Polynomials
Quick Review
Exercises
Divide using long division. Check your answers.
You can divide a polynomial by one of its factors to find
another factor. When you divide by a linear factor, you can
3
+ 7x
2
+ 15x + 9) , (x + 1)
37. (x
simplify this division by writing only the coefficients of each
3
- 7x
2
- 7x + 13) , (x - 4)
38. (2x
term. This is called synthetic division. The Remainder
Theorem says that P (a) is the remainder when you divide
Determine whether each binomial is a factor of
P (x) by x - a.
x
3
+ x
2
− 10x + 8.
39. x - 2
40. x - 4
Example
Let P (x) = 3x
2
- 13x + 15. What is P (3)?
Divide using synthetic division.
According to the Remainder Theorem, P (3) is the
3
+ 5x
2
- x - 5) , (x + 5)
41. (x
remainder when you divide P (x) by x - 3.
3
+ 14x
2
- 58x) , (x + 10)
42. (2x
3
3
13
15
Put the opposite of the
3
+ 8x
2
- 60) , (x - 2)
43. (5x
constant in the divisor
9
12
at the top left.
Use the Remainder Theorem to determine the value
3
4
3
of P (a).
The quotient is 3x - 4 with remainder 3, so P (3) = 3.
44. P (x) = 2x
3
+ 5x
2
+ 7x - 4, a = -2
45. P (x) = x
3
- 4x
2
+ 2x + 3, a = 1
381
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