Practice Makes Perfect Algebra - Carolyn Wheater [19]
6x2 + 5x − 21 =(2x 3)(3x 7)
When you try to place the + and the −, don’t just look at 3 and 7. Instead, look at the inner and the outer. The inner is 9x and the outer is 14x. You want the larger one, 14x, to be positive, so put the + on the 7 and the − on the 3.
6x2 + 5x − 21 =(2x − 3)(3x + 7)
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EXERCISE 7.3
Factor each polynomial.
1. 3x2 + 11x + 10
2. 2x2 − 3x + 1
3. 2x2 + 7x + 3
4. 12x2 + 32x + 5
5. 6x2 + 17x + 12
6. 10x2 − 49x − 5
7. 9x2 − 27x + 20
8. 18x2 + 15x + 2
9. 15x2 − 13x − 6
10. 4x2 − 29x + 30
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Special factoring patterns
Most of the time, factoring is a trial-and-error process, but there are a few cases where the problem is unusual and memorizing is a much better tactic. There are two factoring patterns that should be memorized.
Difference of squares
Most of the expressions you’re asked to factor are trinomials, so you might not expect a binomial like x2 − 4 or 9t2 − 16 be factorable. It turns out that when you multiply the sum and difference of the same two terms with the FOIL rule, however, the outer and inner terms add to 0 and you produce a square minus a square. The difference of squares, a2 − b2, factors to(a + b)(a − b), so x2 − 4 =(x + 2)(x − 2) and 9t2 − 16 =(3t + 4)(3t − 4).
Perfect square trinomial
The perfect square trinomial is one you could figure out how to factor without any memorizing, but it’s convenient to have it memorized to save time. When you square a binomial like 2x + 5, the first and last terms are squares and the inner and outer are identical.
So(ax + b)2 =(ax)2 + 2abx + b2. When you see that the first and last terms of a trinomial are perfect squares, check the middle term to see if you have a perfect square trinomial.
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EXERCISE 7.4
Factor each polynomial.
1. x2 − 49
2. x2 + 6x + 9
3. 36t2 − 1
4. 9t2 − 24t + 16
5. 16 − y2
6. 9y2 + 42y + 49
7. 4x2 − 81
8. 4x2 + 4x + 1
9. 16a2 − 9y2
10. x2 − 12x + 36
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Radicals
If, for some numbers a and b, a2 = b, then a is the square root of b. Seven is the square root of 49 because 72 = 49. Square roots are written using a radical sign: or, in general, . The expression under the radical sign is called the radicand.
You can define other roots in a similar fashion using other powers. Because 23 is equal to 8, 2 is the cube root of 8. Roots other than the square root are indicated by placing a small number, called the index, in the crook of the radical sign, for example, . When no index is shown, the square root is assumed.
When the index of the radical is even, as in square roots, there are both positive and negative roots. Seven is the square root of 49 because 72 = 49, but(−7)2 = 49 as well, so 49 has two square roots, 7 and −7. The positive square root is considered the principal square root, and we agree that will denote the principal root. We’ll write − if we want the negative square root or if we want both.
Simplifying radical expressions
To make radicals easier to work with, put them in simplest radical form. Simplest radical form means the expression contains only one radical, the radicand is as small as possible, and there is no radical in the denominator(or divisor) of a quotient.
Simplest radical form
The principal rule for simplifying radicals tells us that . This rule lets you turn the product of two radicals into a single radical, and it lets you rewrite a radical as a multiple of a smaller radical. You can rewrite as , or simply 6, and you can take and realize that it’s equivalent to , which is equal to because the square root of 4 is 2. By looking for perfect square factors of the radicand and applying this rule, you can rewrite radical expressions with smaller radicands.
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EXERCISE