Question

Find the H.C.F of $ x^2+3x+2$ and $x^3+9x^2+23x+15$.

     (a)X + 1
      (b)X + 2

     (c) $\left(x+1\right)\left(x+2\right)$

      (d)$ \left(x+1\right)\left(x-1\right)$

Answer 1

Hint: In this question we have to find the factor of both the equations. Then find the highest common factor between both factors of the given equations

Complete step-by-step answer:
Step 1 : First step is to find the factors of $x^2+3x+2$. In order to find the factors we have to find the 2 no whose sum is 3 and their product is 2. Those two numbers are 2 and 1 because if we multiply both of the numbers we get 2 and if we add them we get 3.
So, $x^2+3x+2$ can be written as
$x^2+3x+2=x^2+2x+x+2$
Taking x common from $x^2 + 2x$ and 1 from x + 2 we get,
$x\left(x+1\right)+1\left(x+2\right)$
Take x + 2 common we get,
$\left(x+1\right)\left(x+2\right)$
Here we get the factors of $x^2+3x+2$.
Step 2: Now we have to find the factors of $x^3+9x^2+23x+15$. For this equation we are going to apply the hit and trial method. First let’s check for x = -1,
So, by putting the value of x in the equation we get,
$\left(-1\right)^3+9\left(-1^2\right)+23\left(-1\right)+15$
By solving this we get 0, which means (x+1) is a factor of $ x^3+9x^2+23x+15$.
Now, we can express the equation as,
$\left(x+1\right)\left(Ax^2+Bx+C\right)$
Multiplying this we have,
$𝐴x^3 + 𝐵x^2 + 𝐶𝑥 + 𝐴x^2 + 𝐵𝑥 + 𝐶 = 𝐴x^3 + (𝐵+𝐴)x^2 + (𝐶+𝐵)x + 𝐶$
Equating coefficients with our original polynomial, we have:
$x^3 : 𝐴$ = 1
$x^2 : 𝐴$ + 𝐵 = 9 ⇒ 𝐵 = 9 − 𝐴 = 8
x : 𝐵 + 𝐶 = 23 ⇒ 𝐶 = 23 − 𝐵 = 15
So, After putting value of $𝑥^3$ in above equation, we can write the equation as :
$\left(x+1\right)\left(x^2+8x+15\right)$
Now to find the factors for $x^2+8x+15$ we have to find the 2 numbers whose sum is 8 and product is 15.
Those 2 numbers are 5 and 3. Now we can write equation as:
$x^2+5x+3x+5$
Take x common from $x^2$ and 5x and 3 from 3x and 15,
we get:
$x\left(x+5\right)+3\left(x+5\right)$
Now, take x + 5 common we get,
$\left(x+5\right)\left(x+3\right)$
So, the final factor we get for $x^3+9x^2+23x+15$ is:
$\left(x+1\right)\left(x+5\right)\left(x+3\right)$
Now, from the factors of $x^2+3x+2$ and $x^3+9x^2+23x+15$. $\left(x+1\right)$ as the highest common factor because $\left(x+1\right)$ appears in both factors maximum times.
Option (a) is correct.

Note: We can do this question with many other methods like using Quadratic formula, completing the square, graphing etc.
Basically factorisation is the easy way to solve these questions but in some questions where we are not able to find the factors we have to use the other methods.