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Question: How do you factor the expression \(3{x^2} + 11x + 6?\)...

How do you factor the expression 3x2+11x+6?3{x^2} + 11x + 6?

Explanation

Solution

In this question we are going to find the factors of the given expression.
In this we are going to factor the expression by splitting the middle term.
Multiply the coefficient of the first term by a constant in the given expression, we get a number and then find two factors for that number whose sum equals the coefficient of the middle term.
Now rewrite the polynomial by splitting the middle term using the two factors that found before and then add the first two terms and last two terms, taking common factors outside from the first and last two terms. Add the four terms of the above step and we get the desired factorization.
Hence we get the required result.

Complete step by step answer: First write the given expression and mark it as(1)\left( 1 \right).
3x2+11x+63{x^2} + 11x + 6 (1)\left( 1 \right)
The given expression is of the quadratic form ax2+bx+c=0a{x^2} + bx + c = 0
Here the first term is 3x23{x^2} and its coefficient is 3.3.
The middle term is 11x11x and its coefficient is 11.11.
The last term is 66 and it is a constant.
First we are going to multiply the coefficient of first term by the last term.
That is, 3×6=183 \times 6 = 18
Next we are going to find factors of 1818 whose sum is equal to 11.11.
9+2=119 + 2 = 11
By splitting the middle term using the factors 99 and 22 in the given expression.
3x2+9x+2x+63{x^2} + 9x + 2x + 6
Taking common factors outside from the two pairs
3x(x+3)+2(x+3)3x\left( {x + 3} \right) + 2(x + 3)
Rewrite in the factored form,
(3x+2)(x+3)\left( {3x + 2} \right)\left( {x + 3} \right)
(3x+2)=0,(x+3)=0\left( {3x + 2} \right) = 0,\left( {x + 3} \right) = 0
3x=−2,x=−33x = - 2,x = - 3
x=−23,x=−3x = \dfrac{{ - 2}}{3},x = - 3
x=−23,−3x = \dfrac{{ - 2}}{3}, - 3
The required factors of the expression 3x2+11x+63{x^2} + 11x + 6 are (3x+2)(x+3)\left( {3x + 2} \right)\left( {x + 3} \right) .

Note:
We can check our factoring by multiplying them all out to see if we get the original expression. If we do, our factoring is correct, otherwise we had to try again.
The following are some of the factoring methods to solve the expression: factoring out the GCF, the sum product pattern, the grouping method, the perfect square trinomial pattern, the difference of square pattern.