Question

How do you factor \[{a^2} + 4a + 3\] ?

Answer 1

Hint: The given expression is a polynomial of degree 2. Instead of ‘x’ as a variable we have ‘a’ as a variable. We can solve this by using factorization methods or by using quadratic formulas. We use quadratic formula if factorization fails. We know that a polynomial equation has exactly as many roots as its degree.

Complete step-by-step answer:
The degree of the equation \[{a^2} + 4a + 3\] is 2, so the number of roots of the given equation is 2.
On comparing the given equation with the standard quadratic equation \[A{a^2} + Ba + C = 0\] . Where ‘A’ and ‘B’ are coefficients of \[{a^2}\] and coefficient of ‘a’ respectively.
We have \[A = 1\] , \[B = 4\] and \[C = 3\] .
We can expand the middle term ‘4a’ as the sum of ‘3a’ and ‘1a’. Because the product of 3 and 1 is 3 and the sum of 3 and 1 is 4.
 \[{a^2} + 4a + 3 = {a^2} + 3a + a + 3\]
On the right hand side of the equation in the first two terms we take ‘a’ as common and in the remaining term we take 1 as common,
  \[ = a(a + 3) + 1(a + 3)\]
Again taking \[(a + 3)\] as common we have,
 \[ = (a + 3)(a + 1)\]
Hence, the factors of \[{a^2} + 4a + 3\] are \[(a + 3)\] and \[(a + 1)\] .
So, the correct answer is “ \[(a + 3)\] and \[(a + 1)\] ”.

Note: We can find the zeros or roots of the given quadratic expression by equating the obtained factors to zero. That is,
 \[(a + 3)(a + 1) = 0\] .
By zero multiplication property we have,
 \[(a + 3) = 0\] and \[(a + 1) = 0\] .
 \[a = - 3\] and \[a = - 1\] . These are the roots of the given problem.
In various fields of mathematics require the point at which the value of a polynomial is zero, those values are called the factors/solution/zeros of the given polynomial. On the x-axis, the value of y is zero so the roots of an equation are the points on the x-axis that is the roots are simply the x-intercepts.