Sure, let's factorize the given polynomials step by step.
### Polynomial 1: [tex]\( x^2 + 5x - 14 \)[/tex]
To factorize [tex]\( x^2 + 5x - 14 \)[/tex], we need to find two numbers that multiply to give the constant term (-14) and add up to give the coefficient of the middle term (5).
Consider the polynomial:
[tex]\[ x^2 + 5x - 14 \][/tex]
We need to find two numbers that satisfy:
[tex]\[ a \times b = -14 \][/tex]
[tex]\[ a + b = 5 \][/tex]
The numbers that meet these conditions are 7 and -2.
So, we can write the polynomial as:
[tex]\[ x^2 + 5x - 14 = (x + 7)(x - 2) \][/tex]
### Polynomial 2: [tex]\( x^2 - 10x + 16 \)[/tex]
To factorize [tex]\( x^2 - 10x + 16 \)[/tex], we need to find two numbers that multiply to give the constant term (16) and add up to give the coefficient of the middle term (-10).
Consider the polynomial:
[tex]\[ x^2 - 10x + 16 \][/tex]
We need to find two numbers that satisfy:
[tex]\[ a \times b = 16 \][/tex]
[tex]\[ a + b = -10 \][/tex]
The numbers that meet these conditions are -2 and -8.
So, we can write the polynomial as:
[tex]\[ x^2 - 10x + 16 = (x - 2)(x - 8) \][/tex]
### Final Factorized Form:
Therefore:
[tex]\[ x^2 + 5x - 14 = (x + 7)(x - 2) \][/tex]
[tex]\[ x^2 - 10x + 16 = (x - 2)(x - 8) \][/tex]
