You can factor the polynomial f(x)=x^2-19x+70 as (x+a(x+b). The values for a and b are _____.
A. a=10, b=-9
B. a=19, b=70
C. a=-5, b=-14
D. a=-16, b=3
E. none of the above.
(x+a)(x+b) is what i meant.. sorry!
ab=70. Which choice does that.
since 70 = 5*14,
(x-5)(x-14)
so, (C)
To find the values of a and b in the factored form of the polynomial f(x), we need to expand the factored form and compare it to the given polynomial.
The factored form of the polynomial f(x) can be written as (x + a)(x + b). Expanding this expression gives:
(x + a)(x + b) = x^2 + bx + ax + ab = x^2 + (a + b)x + ab
Comparing this to the given polynomial f(x) = x^2 - 19x + 70, we can see that the corresponding coefficients must be the same.
From the comparison, we can conclude that:
Coefficient of x^2: 1 = 1 (since they are the same)
Coefficient of x: -19 = a + b
Constant term: 70 = ab
Now, we need to find the values of a and b that satisfy these equations.
To solve for a and b, we can try each answer choice by substituting the values of a and b into the equations and checking if they satisfy all the conditions.
Let's try answer choice A:
a = 10 and b = -9
Coefficient of x: 10 + (-9) = 1 (satisfies the equation)
Constant term: 10 * (-9) = -90 (does not satisfy the equation)
Since answer choice A does not satisfy all the conditions, we can eliminate it.
Let's try answer choice B:
a = 19 and b = 70
Coefficient of x: 19 + 70 = 89 (does not satisfy the equation)
Since answer choice B does not satisfy all the conditions, we can eliminate it as well.
Let's try answer choice C:
a = -5 and b = -14
Coefficient of x: -5 + (-14) = -19 (satisfies the equation)
Constant term: -5 * (-14) = 70 (satisfies the equation)
Answer choice C satisfies all the conditions, so the values for a and b are a = -5 and b = -14.
Therefore, the correct answer is C.