Which of the binomials below is a factor of this trinomial? x2 – 5x + 4
The Correct Answer and Explanation is:
To factor the trinomial:
x2−5x+4x^2 – 5x + 4×2−5x+4
We are looking for two binomials of the form:(x−a)(x−b)(x – a)(x – b)(x−a)(x−b)
such that when multiplied:
- a+b=5a + b = 5a+b=5
- ab=4ab = 4ab=4
Step 1: Identify two numbers
We need two numbers that add to 5 and multiply to 4. However, since the middle term is -5x, we are actually looking for two numbers that:
- add to -5
- multiply to +4
Try possible factor pairs of 4:
- −1-1−1 and −4-4−4: −1+(−4)=−5-1 + (-4) = -5−1+(−4)=−5, −1×−4=4-1 \times -4 = 4−1×−4=4
Perfect match.
Step 2: Write in factored form
x2−5x+4=(x−1)(x−4)x^2 – 5x + 4 = (x – 1)(x – 4)x2−5x+4=(x−1)(x−4)
So, the binomials that are factors of the trinomial are:
- (x−1)(x – 1)(x−1)
- (x−4)(x – 4)(x−4)
Final Answer:
One binomial factor of x2−5x+4x^2 – 5x + 4×2−5x+4 is: (x−1)\boxed{(x – 1)}(x−1)
Explanation:
Factoring a trinomial involves breaking it into the product of two binomials. The standard form is:x2+bx+cx^2 + bx + cx2+bx+c
To factor it, find two numbers whose sum is b and product is c. In this case, b=−5b = -5b=−5, c=4c = 4c=4. The numbers −1-1−1 and −4-4−4 meet these conditions. When you expand (x−1)(x−4)(x – 1)(x – 4)(x−1)(x−4) using the FOIL method:
- First: x×x=x2x \times x = x^2x×x=x2
- Outer: x×−4=−4xx \times -4 = -4xx×−4=−4x
- Inner: −1×x=−1x-1 \times x = -1x−1×x=−1x
- Last: −1×−4=4-1 \times -4 = 4−1×−4=4
Adding these: x2−4x−x+4=x2−5x+4x^2 – 4x – x + 4 = x^2 – 5x + 4×2−4x−x+4=x2−5x+4
So the factorization is correct. Therefore, (x−1)(x – 1)(x−1) is one of the binomial factors.
