a 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.
6x^2+5x-6
⇔
6x^2+5x+(- 6)
We have that a= 6, b=5, and c=- 6. There are now three steps we need to follow in order to rewrite the above expression.
Find a c. Since we have that a= 6 and c=- 6, the value of a c is 6* (- 6)=- 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 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, so that their sum is positive.
b We want to factor the given equation. Let's start factoring by first identifying the greatest common factor (GCF).
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. In this case, the GCF is 2.
The result of factoring out a GCF from the given expression is a binomial with a leading coefficient of 4.
2( 4x^2-25)
Let's temporarily only focus on this binomial, and we will bring back the GCF after factoring.
Factor the Expression
Do you notice that the binomial is a difference of two perfect squares? This can be factored using the difference of squares method.
a^2 - b^2 ⇔ (a+b)(a-b)
To do so, we first need to express each term as a perfect square.
Expression
4x^2-25
Rewrite as Perfect Squares
(2x)^2 - 5^2
Apply the Formula
(2x+5)(2x-5)
Wait! Before we finish, remember that we factored out a GCF from the original expression. To fully complete the factored expression, let's reintroduce that GCF now.
2(2x+5)(2x-5)
c Let's start factoring by first identifying the GCF. Then, we will rewrite the expression as a trinomial with a leading coefficient of 1.
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. In this case, the GCF is 2x.
The result of factoring out a GCF from the given expression is a trinomial with a leading coefficient of 1.
2( x^2+x-56)
Let's temporarily only focus on this trinomial, and we will bring back the GCF after factoring.
Factor the Expression
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.
x^2+x- 56
In this case we have - 56. This is a negative number, so for the product of the constant terms in the factors to be negative, these constants must have the opposite sign — one positive and one negative.
Factor Constants
Product of Constants
1 and - 56
- 56
-1 and 56
- 56
2 and - 28
- 56
-2 and 28
- 56
4 and - 14
- 56
-4 and 14
- 56
7 and - 8
- 56
- 7 and 8
- 56
Next, let's consider the coefficient of the linear term.
x^2+1x- 56
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 - 56
- 55
-1 and 56
55
2 and - 28
- 26
-2 and 28
26
4 and - 14
- 10
-4 and 14
10
7 and - 8
- 1
-7 and 8
1
We found the factors whose product is - 56 and whose sum is 1.
x^2+1x- 56 ⇔ (x-7)(x+8)
Wait! Before we finish, remember that we factored out a GCF from the original expression. To fully complete the factored expression, let's reintroduce that GCF now.
2x(x-7)(x+8)
d We want to factor the given expression. Factoring is much easier when our polynomial is a perfect square trinomial. To determine if an expression is a perfect square trinomial, we need to ask ourselves three questions.
Is the first term a perfect square?
9x^2=( 3x)^2 ✓
Is the last term a perfect square?
16= 4^2 ✓
Is the middle term twice the product of 4 and 3x?
24x=2* 4* 3x ✓
As we can see, the answer to all three questions is yes! Therefore, we can write the trinomial as the square of a binomial. Note there is a subtraction sign in the middle.
9x^2-24x+16 ⇔ ( 3x- 4)^2
