Chapter Closure
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Exercise 107 Page 363
a Is there a greatest common factor (GCF) between all of the terms in the given expression? If so, you should factor that out first.
b Is there a GCF between all of the terms in the given expression? If so, you should factor that out first.
c Is there a GCF between all of the terms in the given expression? If so, you should factor that out first.
d Is there a GCF between all of the terms in the given expression? If so, you should factor that out first.
a (x-5)(x+6)
b x(x-7)(- 3x+2)
c Not factorable, see solution.
d 2x(x+3)(3x-4)
a To factor a trinomial with a leading coefficient of 1, think of the process as multiplying two binomials in reverse. Let's start by taking a look at the constant term.
| Factor Constants | Product of Constants |
|---|---|
| 1 and - 30 | - 30 |
| -1 and 30 | - 30 |
| 2 and - 15 | - 30 |
| -2 and 15 | - 30 |
| 3 and - 10 | - 30 |
| -3 and 10 | - 30 |
| 5 and - 6 | - 30 |
| -5 and 6 | - 30 |
Next, let's consider the coefficient of the linear term. x^2+1x- 30 For this term we need the sum of the factors that produced the constant term to equal the coefficient of the linear term, 1.
| Factors | Sum of Factors |
|---|---|
| 1 and - 30 | - 29 |
| -1 and 30 | 29 |
| 2 and - 15 | - 13 |
| -2 and 15 | 13 |
| 3 and - 10 | - 7 |
| -3 and 10 | 7 |
| 5 and - 6 | - 1 |
| -5 and 6 | 1 |
We found the factors whose product is - 30 and whose sum is 1. x^2+1x- 30 ⇔ (x-5)(x+6)
b We want to completely factor the given expression. To do so, we will first identify and factor out the greatest common factor.
Factor Out the GCF
The greatest common factor (GCF) of an expression is a common factor of the terms in the expression. It is the common factor with the greatest coefficient and the greatest exponent. The GCF of the given expression is x.- 3x^3+23x^2-14x
SplitIntoFactors
Split into factors
x(- 3x^2)+ x(23x)- x(14)
FactorOut
Factor out x
x(- 3x^2+23x-14)
Factor the Quadratic Trinomial
Here we have a quadratic trinomial of the form ax^2+bx+c, where |a| ≠ 1 and there are no common factors. To factor this expression we will rewrite the middle term, bx, as two terms. The coefficients of these two terms will be factors of ac whose sum must be b. - 3x^3+23x^2-14x ⇕ x ( - 3x^2+23x+(- 14) ) We have that a= - 3, b=23, and c=- 14. There are now three steps we need to follow in order to rewrite the above expression.
- Find a c. Since we have that a= - 3 and c=- 14, the value of a c is - 3* (- 14)=42.
- Find factors of a c. Since ac=42, which is positive, we need factors of a c to have the same sign — both positive or both negative — in order for the product to be positive. Since b=23, which is also positive, those factors will need to be positive so that their sum is positive.
c|c|c|c 1^(st)Factor &2^(nd)Factor &Sum &Result 1 &42 &1 + 42 &43 2 & 21 & 2 + 21 &23 3 &14 &3+14 &17 6 &7 &6 + 7 &13
- Rewrite bx as two terms. Now that we know which factors are the ones to be used, we can rewrite bx as two terms. x (- 3x^2+23x- 14 ) ⇕ x (- 3x^2+ 2x+ 21x-14 )
c Here we have a quadratic trinomial of the form ax^2+bx+c, where |a| ≠ 1 and there are no common factors. To factor this expression we will try to rewrite the middle term, bx, as two terms. The coefficients of these two terms would have to be factors of ac whose sum must be b.
We have that a= 2, b=- 5, and c=4. There are now a few steps we need to follow in order to rewrite the above expression.
- Find a c. Since we have that a= 2 and c=4, the value of a c is 2* 4=8.
- Find factors of a c. Since a c=8, which is positive, we need factors of a c to have the same sign — both positive or both negative — in order for the product to be positive. Since b=- 5, which is negative, those factors will need to be negative so that their sum is negative.
c|c|c|c 1^(st)Factor &2^(nd)Factor &Sum &Result - 1 &- 8 &- 1 + (- 8) &- 9 - 2 &- 4 &- 2 + (- 4) &- 6 We cannot find a pair of integers whose product is 8 and whose sum is - 5. Therefore, we cannot use the general method for factoring trinomials. The given polynomial is not factorable.
d We want to completely factor the given expression. To do so, we will first identify and factor out the greatest common factor.
Factor Out the GCF
The GCF of an expression is a common factor of the terms in the expression. It is the common factor with the greatest coefficient and the greatest exponent. The GCF of the given expression is 2x.6x^3+10x^2-24x
SplitIntoFactors
Split into factors
2x(3x^2)+ 2x(5x)- 2x(12)
FactorOut
Factor out 2x
2x(3x^2+5x-12)
Factor the Quadratic Trinomial
Here we have a quadratic trinomial of the form ax^2+bx+c, where |a| ≠ 1 and there are no common factors. To factor this expression, we will rewrite the middle term, bx, as two terms. The coefficients of these two terms will be factors of ac whose sum must be b. 2x(3x^2+5x-12) ⇕ 2x( 3x^2+5x+(- 12)) We have that a= 3, b=5, and c=- 12. There are now three steps we need to follow in order to rewrite the above expression.
- Find a c. Since we have that a= 3 and c=- 12, the value of a c is 3* (- 12)=- 36.
- Find factors of a c. Since ac=- 36, which is negative, we need factors of a c to have opposite signs — one positive and one negative. Since their sum b=5, which is positive, the absolute value of the positive factor will need to be greater than the absolute value of the negative factor.
c|c|c|c 1^(st)Factor &2^(nd)Factor &Sum &Result - 1 &36 &-1 + 36 &35 - 2 &18 &-2+18 &16 - 3 &12 &-3+12 &9 - 4 & 9 & - 4 + 9 &5 - 6 &6 &-6 + 6 &0
- Rewrite bx as two terms. Now that we know which factors are the ones to be used, we can rewrite bx as two terms. 2x (3x^2+5x-12) ⇕ 2x (3x^2 - 4x+ 9x-12)
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