Question

Factorize
(1) $ {x^2} + 9x + 18 $
(2) $ {x^2} - 10x + 9 $
(3) $ {y^2} + 24y + 144 $
(4) $ 5{y^2} + 5y - 10 $
(5) $ {p^2} - 2p - 35 $
(6) $ {p^2} - 7p - 44 $
(7) $ {m^2} - 23m + 120 $
(8) $ {m^2} - 25m + 100 $
(9) $ 3{x^2} + 14x + 15 $
(10) $ 2{x^2} + x - 45 $
(11) $ 20{x^2} - 26x + 8 $ (12) $ 44{x^2} - x - 3 $

Answer 1

Hint: Every quadratic polynomial will have two factors. As the above 12 polynomials are quadratic, each polynomial will have two factors. First step is to divide the middle term into two according to the product of coefficients of first and third terms.

Complete step-by-step answer:
We are given to factorize quadratic polynomials. Factorization of a polynomial is the process of writing the polynomial in terms of its factors.
(1) $ {x^2} + 9x + 18 $
 $ {x^2} + 9x + 18 $ can also be written as
 $
  {x^2} + 3x + 6x + 18 \\
  \Rightarrow x\left( {x + 3} \right) + 6\left( {x + 3} \right) \\
  \Rightarrow \left( {x + 3} \right)\left( {x + 6} \right) \;
  $
Therefore, the factors of $ {x^2} + 9x + 18 $ are $ \left( {x + 3} \right),\left( {x + 6} \right) $

(2) $ {x^2} - 10x + 9 $
 $ {x^2} - 10x + 9 $ can also be written as
 $
  {x^2} - 9x - x + 9 \\
  \Rightarrow x\left( {x - 9} \right) - 1\left( {x - 9} \right) \\
  \Rightarrow \left( {x - 9} \right)\left( {x - 1} \right) \;
  $
Therefore, the factors of $ {x^2} - 10x + 9 $ are $ \left( {x - 9} \right),\left( {x - 1} \right) $

(3) $ {y^2} + 24y + 144 $
 $ {y^2} + 24y + 144 $ can also be written as
 $
  {y^2} + 12y + 12y + 144 \\
  \Rightarrow y\left( {y + 12} \right) + 12\left( {y + 12} \right) \\
  \Rightarrow \left( {y + 12} \right)\left( {y + 12} \right) = {\left( {y + 12} \right)^2} \;
  $
 $ {y^2} + 24y + 144 $ is $ {\left( {y + 12} \right)^2} $

(4) $ 5{y^2} + 5y - 10 $
 $ 5{y^2} + 5y - 10 $ can also be written as
 $
  5{y^2} + 10y - 5y - 10 \\
  \Rightarrow 5y\left( {y + 2} \right) - 5\left( {y + 2} \right) \\
  \Rightarrow \left( {y + 2} \right)\left( {5y - 5} \right) \;
  $
The factors of $ 5{y^2} + 5y - 10 $ are \[\left( {y + 2} \right),\left( {5y - 5} \right)\]

(5) $ {p^2} - 2p - 35 $
 $ {p^2} - 2p - 35 $ can also be written as
 $
  {p^2} - 7p + 5p - 35 \\
  \Rightarrow p\left( {p - 7} \right) + 5\left( {p - 7} \right) \\
  \Rightarrow \left( {p - 7} \right)\left( {p + 5} \right) \;
  $
Therefore, the factors of $ {p^2} - 2p - 35 $ are $ \left( {p - 7} \right),\left( {p + 5} \right) $

(6) $ {p^2} - 7p - 44 $
 $ {p^2} - 7p - 44 $ can also be written as
 $
  {p^2} - 11p + 4p - 44 \\
  \Rightarrow p\left( {p - 11} \right) + 4\left( {p - 11} \right) \\
  \Rightarrow \left( {p - 11} \right)\left( {p + 4} \right) \;
  $
Therefore, the factors of $ {p^2} - 7p - 44 $ are $ \left( {p - 11} \right),\left( {p + 4} \right) $

(7) $ {m^2} - 23m + 120 $
 $ {m^2} - 23m + 120 $ can also be written as
 $
  {m^2} - 15m - 8m + 120 \\
  \Rightarrow m\left( {m - 15} \right) - 8\left( {m - 15} \right) \\
  \Rightarrow \left( {m - 15} \right)\left( {m - 8} \right) \;
  $
The factors of $ {m^2} - 23m + 120 $ are $ \left( {m - 15} \right),\left( {m - 8} \right) $

(8) $ {m^2} - 25m + 100 $
 $ {m^2} - 25m + 100 $ can also be written as
 $
  {m^2} - 20m - 5m + 100 \\
  \Rightarrow m\left( {m - 20} \right) - 5\left( {m - 20} \right) \\
  \Rightarrow \left( {m - 20} \right)\left( {m - 5} \right) \;
  $
The factors of $ {m^2} - 25m + 100 $ are $ \left( {m - 20} \right),\left( {m - 5} \right) $

(9) $ 3{x^2} + 14x + 15 $
  $ 3{x^2} + 14x + 15 $ can also be written as
 $
  3{x^2} + 9x + 5x + 15 \\
  \Rightarrow 3x\left( {x + 3} \right) + 5\left( {x + 3} \right) \\
  \Rightarrow \left( {x + 3} \right)\left( {3x + 5} \right) \;
  $
The factors of $ 3{x^2} + 14x + 15 $ are $ \left( {x + 3} \right),\left( {3x + 5} \right) $

(10) $ 2{x^2} + x - 45 $
 $ 2{x^2} + x - 45 $ can also be written as
 $
  2{x^2} + 10x - 9x - 45 \\
  \Rightarrow 2x\left( {x + 5} \right) - 9\left( {x + 5} \right) \\
  \Rightarrow \left( {x + 5} \right)\left( {2x - 9} \right) \;
  $
The factors of $ 2{x^2} + x - 45 $ are $ \left( {x + 5} \right),\left( {2x - 9} \right) $

(11) $ 20{x^2} - 26x + 8 $
 $ 20{x^2} - 26x + 8 $ can also be written as
 $
  20{x^2} - 10x - 16x + 8 \\
  \Rightarrow 10x\left( {2x - 1} \right) - 8\left( {2x - 1} \right) \\
  \Rightarrow \left( {2x - 1} \right)\left( {10x - 8} \right) \;
  $
The factors of $ 20{x^2} - 26x + 8 $ are $ \left( {2x - 1} \right),\left( {10x - 8} \right) $

(12) $ 44{x^2} - x - 3 $
 $ 44{x^2} - x - 3 $ can also be written as
 $
  44{x^2} - 11x + 12x - 3 \\
  \Rightarrow 11x\left( {4x - 1} \right) + 3\left( {4x - 1} \right) \\
  \Rightarrow \left( {4x - 1} \right)\left( {11x + 3} \right) \;
  $
The factors of $ 44{x^2} - x - 3 $ are $ \left( {4x - 1} \right),\left( {11x + 3} \right) $

Note: When we are factoring a quadratic polynomial, always remember that the middle term is divided into two in such a way that the product of the coefficients of divided terms must be equal to the product of the coefficients of first and last terms. Otherwise we would not be able to factorize the polynomial. And be careful with the signs of the terms.