Introduction
In elementary algebra, a trinomial is a polynomial consisting of three terms or monomials. In mathematics, factorization (also factorisation in British English) or factoring is the decomposition of an object (for example, a number, a polynomial, or a matrix) into a product of other objects, or factors, which when multiplied together give the original. In this article we shall discuss about factoring quadratic trinomial.
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Example Problem on Factoring Trinomials
Problem 1:
Factoring trinomials
15 – 2x – x2.
Solution: Writing in the standard form,
15 – 2x – x2 = –x2 – 2x + 15
= (–1) (x2 + 2x – 15).
Here, we find –15 = 5 × –3, 5 + (–3) = 2
Hence, we get 15 – 2x – x2 = (–1) [(x+5) {x + (–3)}]
= (–1) (x +5)(x – 3)
= (x + 5) ( 3 – x).
Problem 2:
Factoring trinomials
x2 – x – 132.
Solution: We find –132 = (–12) × (11), (–12) + 11 = –1.
Hence we get x2 – x – 132 = [x + (–12)] (x + 11) = (x – 12) (x + 11).
Next, we consider the quadratic polynomial ax2 + bx + c where a, b, c are integers and a ≠0,1. If we are able to find two integers p and q such that pq = ac and p + q = b. Then
ax2 + bx + c =`1/a ` (a2x2 + abx + ac)
=`1/a` [a2x2 + a(p+q)x + pq]= `1/a` [a2x2 + apx + aqx + pq]=`1/a` [ax (ax + p) + q(ax + p)]
=`1/a` (ax + p) (ax + q)
Thus, we are able to factorize the expression
Problem 3:
Factoring trinomial
2x2+ 7x + 3.
Solution: Here a = coefficient of x2 = 2
b = coefficient of x = 7
c = constant term = 3
We find a × c = 2 × 3 = 6 = 6 × 1, 6 + 1 = 7 = b. Hence
2x2 + 7x + 3 = 21 (2x + 6) (2x + 1) =(x+3)(2x+1).
Instead of applying the final result of the rule, we can also do the factorization by splitting the middle term and grouping as follows:
2x2 + 7x + 3 = 2x2 + (6 + 1)x + 3
= 2x2 + 6x + x + 3
= 2x(x + 3) + (1)(x+3) = (2x+1) (x+3).
Practice Problem on Factoring Quadratic Trinomials
Problem 1:
Factorize 8a2 + 2a – 3.
Answer:
(4a + 3) (2a – 1)
Problem 2:
Factorize 6 + 11/2 x + x2.
Answer:
1/2 (x + 4) (2x + 3).
In elementary algebra, a trinomial is a polynomial consisting of three terms or monomials. In mathematics, factorization (also factorisation in British English) or factoring is the decomposition of an object (for example, a number, a polynomial, or a matrix) into a product of other objects, or factors, which when multiplied together give the original. In this article we shall discuss about factoring quadratic trinomial.
My forthcoming post is on math word problems 6th grade, math problem solver with steps free will give you more understanding about Algebra
Example Problem on Factoring Trinomials
Problem 1:
Factoring trinomials
15 – 2x – x2.
Solution: Writing in the standard form,
15 – 2x – x2 = –x2 – 2x + 15
= (–1) (x2 + 2x – 15).
Here, we find –15 = 5 × –3, 5 + (–3) = 2
Hence, we get 15 – 2x – x2 = (–1) [(x+5) {x + (–3)}]
= (–1) (x +5)(x – 3)
= (x + 5) ( 3 – x).
Problem 2:
Factoring trinomials
x2 – x – 132.
Solution: We find –132 = (–12) × (11), (–12) + 11 = –1.
Hence we get x2 – x – 132 = [x + (–12)] (x + 11) = (x – 12) (x + 11).
Next, we consider the quadratic polynomial ax2 + bx + c where a, b, c are integers and a ≠0,1. If we are able to find two integers p and q such that pq = ac and p + q = b. Then
ax2 + bx + c =`1/a ` (a2x2 + abx + ac)
=`1/a` [a2x2 + a(p+q)x + pq]= `1/a` [a2x2 + apx + aqx + pq]=`1/a` [ax (ax + p) + q(ax + p)]
=`1/a` (ax + p) (ax + q)
Thus, we are able to factorize the expression
Problem 3:
Factoring trinomial
2x2+ 7x + 3.
Solution: Here a = coefficient of x2 = 2
b = coefficient of x = 7
c = constant term = 3
We find a × c = 2 × 3 = 6 = 6 × 1, 6 + 1 = 7 = b. Hence
2x2 + 7x + 3 = 21 (2x + 6) (2x + 1) =(x+3)(2x+1).
Instead of applying the final result of the rule, we can also do the factorization by splitting the middle term and grouping as follows:
2x2 + 7x + 3 = 2x2 + (6 + 1)x + 3
= 2x2 + 6x + x + 3
= 2x(x + 3) + (1)(x+3) = (2x+1) (x+3).
Practice Problem on Factoring Quadratic Trinomials
Problem 1:
Factorize 8a2 + 2a – 3.
Answer:
(4a + 3) (2a – 1)
Problem 2:
Factorize 6 + 11/2 x + x2.
Answer:
1/2 (x + 4) (2x + 3).
