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Factoring trinomials formula is an integration of different algebraic terms such as factoring and trinomials. This formula is required for solving complex problems which deal with polynomials. Trinomial is a mathematical algebraic expression that consists of three terms ax2+bx+c. In other words, trinomial is the polynomial expression in the form of ax²+bx+c. An expression consisting of one term or more with a non-zero coefficient (Variables with non-negative exponents) is known as a polynomial.
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Keyterms: Factor, Trinomial, Integration, Algebric Terms, Exponents, Mathematical expression, Graphic functions, Factoring
Trinomial Terms
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Factoring trinomial formula is an advanced level of mathematical expression which is associated with different algebraic terms. It is useful to recall some of the terms associated with the factoring trinomial formula.
Trinomials
Trinomial is an expression containing three terms. Articulating an expression as the product of more than one expression is called factoring. Factoring is engaged at every algebra level for solving polynomials, simplifying complex expressions, and for solving graphic functions. In general terms, factoring is the inverse application of expanding an expression.
Like 3(x-2) is a factored form of 3x-6 and (x-1) (x-6) is a factored form of x2 + 5x − 6. Factoring is a little challenging but expanding is an approximately straightforward activity. Hence, every student must practice different types of factorization in order to gain proficiency in applying them.
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- Factors: A factor divides another number without leaving a reminder. Each number consists of a factor which is less than or equal to the number. The factors of 12 are 1, 2, 3, 4, 6, and 12 themselves. It can be concluded that all numbers have a factor of 1 and all numbers are a factor of itself.
- Factoring: It helps to rewrite a polynomial into simple factors. By equating the factors to zero, one can understand the solutions of any given polynomial equation.
- Common Factors: Common factors can be divided into two or more numbers beyond leaving a reminder. The common factors of numbers 60, 90, and 150 are 1, 2, 3, 5, 6, 10, 15, and 30.
- GCF: Greatest Common Factor (GCF) is the biggest value of factors of the given numbers. Like the GCF of 60, 90, and 150 are 1, 2, 3, 5, 6, 10, 15, and 30. Hence the largest common factor is 30.
- Polynomial: It is an algebraic expression with more than two terms such as numbers and variables, commonly combined by addition or subtraction operations. Some of the polynomials are 3x + 4xy – 5y, 2x + 3, 3xy – 4y, and x2 − 4x + 7.
- Trinomial: It is an algebraic formula made of three terms, normally in the form ax2 + bx + c = 0, where a, b, and c are numerical coefficients. The ‘a’ is known as the leading coefficient and it is not equal to zero (a≠0).
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Factoring Trinomials Formula
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Factoring trinomials formula is required to solve complex mathematical problems involving polynomials. The trinomial could be a perfect square or a non-perfect square. There are 2 formulas to factorize a perfect square trinomial. But for a non-perfect square trinomial, there is no particular formula available. Instead, there is a process which is given below.
Factoring trinomials formula of perfect trinomials are as follows
a2 + 2ab + b2 = (a + b)2
a2 - 2ab + b2 = (a - b)2
In order to apply either of these formulas, the trinomial must be a2 + 2ab + b2 (or) a2 - 2ab + b2.
The step by step procedure of factoring a non-perfect trinomial ax2 + bx + c is given below:
- Step1: Identify ac and b.
- Step2: Identify the numbers whose product is ac and sum is b..
- Step3: In this step, one must split the middle term as the sum of two terms with the numbers obtained from step2.
- Step4: Factor by grouping
In order to factorize a trinomial of the form ax2 + bx + c, we can use any of the formulas:
- a2 + 2ab + b2 = (a + b)2 = (a + b) (a + b)
- a2 - 2ab + b2 = (a - b)2 = (a - b) (a - b)
- a2 - b2 = (a + b) (a - b)
- a3 + b3 = (a + b) (a2 - ab + b2)
- a3 - b3 = (a - b) (a2 + ab + b2)
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Things to Remember
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- Factoring trinomials formula is a fundamental formula used for solving complex problems which deal with polynomials.
- Trinomials is a polynomial expression in the form of ax2+bx+c.
- The general term of Factoring means the inverse application of expanding an expression.
- Polynomial is an algebraic expression with more than two terms such as numbers and variables, commonly combined by addition or subtraction operations. Examples of the polynomials are 3x + 4xy – 5y, 2x + 3, 3xy – 4y, and x2 − 4x + 7.
Sample Questions
Ques: Factor x2 - 16x + 64 with factoring trinomials formulas. (2 marks)
Ans: Let's rewrite the given formula as:
x2 - 16x + 64 = x2 - 2 (x) (8) + 82
The right side trinomial is in this form
a2 - 2ab + b2
hence the formula can be applied
a2 - 2ab + b2 = (a - b)2
Thus, x2 - 2 (x) (8) + 82 = (x - 8)2
= x2 - 16x + 64 = (x - 8)2.
Ques: Factor the trinomial 2x2 - x - 3. (3 marks)
Ans: We can factor the given trinomial using the factoring trinomial formula of non-perfect trinomials
Compare 2x2 - x - 3 with ax2 + bx + c
a = 2, b = -1, and c = -3
so, ac = 2 (-3) = -6 and b = -1
numbers whose product is -6 and sum is -1 are -3 and 2
let’s split the middle term -x as -3x + 2x and factor by grouping the terms
2x2 - x - 3
= 2x2 - 3x + 2x - 3
= x (2x - 3) + 1 (2x - 3)
= (2x - 3)(x + 1)
Hence the solution is = 2x2 - x - 3 = (2x - 3) (x + 1).
Ques: What are the formulas we can use in order to factorize a trinomial of the form ax2 + bx + c? (2 marks)
Ans: We can use all the given formulas to factorize a trinomial of the form ax2 + bx + c.
- a2 + 2ab + b2 = (a + b)2 = (a + b) (a + b)
- a2 - 2ab + b2 = (a - b)2 = (a - b) (a - b)
- a2 - b2 = (a + b) (a - b)
- a3 + b3 = (a + b) (a2 - ab + b2)
- a3 - b3 = (a - b) (a2 + ab + b2)
Ques: If y-3 is a factor of y2+a - 6y, then determine the value of a. Also, find the other factor of the trinomial. (3 marks)
Ans: (y - 3) is a factor of y2 + a - 6y. If we put (y = 3) in the trinomial y2 + a - 6y, then the value will be 0.
32+a 6×3 = 0
9+a-18 = 0
a-9 = 0
a=9
factorize the trinomial y2 + a - 6y = y2 + 9 - 6y
The given trinomial is the expansion of the identity (x - y)2 = x2 - 2xy + y2
y2-6y+9
= y2 - 2 × 3 × y + 32 = (y - 3)2
hence, a=9 and y2+a-6y
= (y - 3)2.
Ques: Find the factors of 15a2 + 38ab + 24b2 (2 marks)
Ans: Break the middle term, multiply the coefficient of a2 and b2 = 15 × 24 = 360
find two numbers where on multiplication they should get 360 and in addition, they should give the result 38.
18 × 20 = 360 and 18 + 20 = 38
15a2 + 38ab + 24b2
= 15a2 + 18ab + 20ab + 24b2
= 3a (5a + 6b) + 4b (5a + 6b)
= (5a + 6b)(3a + 4b)
Therefore, (5a + 6b)(3a + 4b) is the factor for 15a2 + 38ab + 24b2.
Ques: Determine the factors of x2 - 5x + 6. (2 marks)
Ans: x2 - 5x + 6
= x2 - 3x - 2x + 3 × 2
= x (x - 3) - 2x + 6
= x (x - 3) - 2(x - 3)
= (x - 2) (x - 3)
Hence, (x - 2) (x - 3) is the factor of x2 - 5x + 6.
Ques: The income of Lara is $ (2x2−4y2+3xy−5) and her expenditure is $ (−2y2+5x2+9). Consider the concept of subtraction of polynomials to find her savings. (3 marks)
Ans: Savings = Income - expenditure
Now, apply the same formula here,
Savings = 2x2−4y2+3xy−5−(9−2y2+5x2)
= 2x2−4y2+3xy−5+2y2−5x2−9
= −3x2−2y2+3xy−14
Hence, her savings is:
= $(−3x2−2y2+3xy−14).
Ques: Using the factoring polynomials techniques, factor x3 + 5x2 + 6x. (3 marks)
Ans: Firstly, reduce the degree of the polynomial from 3 to 2.
Here, x is a common factor in x3 + 5x2 + 6x.
so, x3 + 5x2 + 6x
= x(x2 + 5x + 6)
Now split x2 +5x+6 as x2 + 3x + 2x + 6
x2 + 5x + 6
= x(x + 3) + 2(x + 3)
= (x + 3) (x + 2)
Hence, on factoring the cubic polynomial x3 + 5x2 + 6x we get the result x(x + 2) (x + 3) as its factors.
So the answer is x3 + 5x2 + 6x
= x(x+2) (x+3).
Ques: Factorize the polynomial 6xy – 4y + 6 – 9x with the method of regrouping or factoring a polynomial. (2 marks)
Ans: For factoring polynomials there are no common factors among all the terms in the expression 6xy-4y+6-9x.
so regroup them as (6xy-4y) and (6-9x)
(6xy - 4y) + (6 - 9x)
= 2y(3x - 2)- 3(3x - 2)
= (3x - 2) (2y - 3)
Hence, on factoring polynomials 6xy - 4y + 6 - 9x, we get (2y - 3) and (3x - 2) as the factors.
Ques: The quadratic equation 2x2 + 9x + 7 = 0 has roots α, β. Identify the quadratic equation having the roots 1/α, and 1/β. (2 marks)?
Ans: The quadratic equation with roots reciprocal to the roots of the equation ax2 + bx + c = 0 and cx2 + bx + a = 0
The given quadratic equation is 2x2 + 9x + 7 = 0
So the required equation with reciprocal roots is 7x2 + 9x + 2 = 0
Hence the equation is 7x2 + 9x + 2 = 0
Ques: Find the quadratic equation with the roots 5 and 8 respectively. (2 marks)
Ans: The quadratic equation with the roots α, β, is x2 - (α + β)x + αβ = 0
Here α = 5, and β = 8
So the quadratic equation is:
x2 - (5 + 8)x + 5×8 = 0
x2 - 13x + 40 = 0
Hence the quadratic equation is x2 - 13x + 40 = 0
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