Please Explain how to factor the following trinomials forms: x2 + bx + c and ax2 + bx + c in words.
for x^2+bx+c you are looking for factors
(x+h)(x+k) = x^2 + (h+k)x + hk
So, you want to find two numbers h and k such that
h+k = b
hk = c
Same thing for the 2nd one, except that now you have to contend with factors of a as well.
To factor trinomials of the form x^2 + bx + c and ax^2 + bx + c, we need to find two binomials whose product equals the given trinomial.
For trinomials of the form x^2 + bx + c:
1. Look at the constant term, c. We need to find two numbers whose product equals c.
2. Look at the coefficient of x, b. We need to find two numbers whose sum equals b.
3. Express the coefficient of x, b, as the sum or difference of the two numbers found in step 2.
4. Rewrite the trinomial, replacing bx with the two numbers from step 3.
5. Factor by grouping or use the distributive property to rewrite the expression as the product of two binomials.
For example, let's factor the trinomial x^2 + 7x + 10:
1. The constant term is 10. We need to find two numbers whose product equals 10.
The possible pairs of numbers are (1, 10), (2, 5), and (-1, -10), (-2, -5).
2. The coefficient of x is 7. We need to find two numbers whose sum equals 7.
The pair of numbers that sums up to 7 is (2, 5).
3. We rewrite 7x as 2x + 5x.
4. The trinomial becomes x^2 + 2x + 5x + 10.
5. We can now factor by grouping: (x^2 + 2x) + (5x + 10) = x(x+2) + 5(x+2).
Factoring out the common binomial, we get (x+2)(x+5), which is the factored form.
For trinomials of the form ax^2 + bx + c:
1. The steps are the same as before, but for this form, we need to take into account the coefficient a in front of x^2.
2. Start by multiplying the coefficient a with the constant term c to find two numbers whose product equals ac.
3. Find two numbers whose sum equals b, just like in the previous form.
4. Rewrite the trinomial, replacing bx with the two numbers found in step 3 and ac with the product from step 2.
5. Factor by grouping or use the distributive property to rewrite the expression as the product of two binomials.
For example, let's factor the trinomial 2x^2 + 7x + 3:
1. The constant term is 3. We need to find two numbers whose product equals 2*3 = 6.
The possible pairs of numbers are (1, 6) and (-1, -6).
2. The coefficient of x is 7. We need to find two numbers whose sum equals 7.
The pair of numbers that sums up to 7 is (1, 6).
3. We rewrite 7x as 1x + 6x.
4. The trinomial becomes 2x^2 + x + 6x + 3.
5. We can now factor by grouping: (2x^2 + x) + (6x + 3) = x(2x+1) + 3(2x+1).
Factoring out the common binomial, we get (2x+1)(x+3), which is the factored form.
Remember to always check your answer by multiplying the two binomials to make sure they equal the original trinomial.