Ok I got two I need help with :
1. Factor Completely
7x^2 + 49x + 42
I get 7x^2 + 7x + 42x + 42
then 7x(x+1) but the right side is confusing me. I know it should be (x+6).
So the answer is 7x(x+1)(x+6) But I get 7x(x+1) 7(6x+6). What am I doing wrong?
2.Factor completley:
x^10y^3 - 4x^9y^2 - 21x^8y
Totally consfused on this one.
On the first, factor 42 out of the second paren.
7x(x+1)+ 42(x+1)
(x+1)(7x+42)
now factor 7 out of the second..
7(x+1)(x+6)
Onthe last, factor the common factor x^8*y out
yx^8[(xy)^2 -4xy -21]
The last term factors to (xy-7)(xy+3)
Solution for No.1
7x^2 + 49x + 42
7x^2 + 7x + 42x + 42
(7x^2 + 7x) + (42x + 42)
[7x(x+1)+ 42(x+1)]
(7x+42)(x+1)
7(x+6)(x+1)
Solution for No.2
x^10y^3 - 4x^9y^2 - 21x^8y
x^8y (x^2y^2 - 4xy - 21)
x^8y (xy-7)(xy+3)
1. To factor the quadratic expression 7x^2 + 49x + 42 completely, you can use the method of factoring by grouping. Here's how you can do it:
Step 1: Multiply the coefficients of the first and the last term, which gives you 7 * 42 = 294.
Step 2: Now, find two numbers that multiply to give 294 and add up to give the coefficient of the middle term (49x). In this case, the numbers are 42 and 7, because 42 * 7 = 294 and 42 + 7 = 49.
Step 3: Rewrite the expression, splitting the middle term:
7x^2 + 42x + 7x + 42
Step 4: Now factor by grouping:
(7x^2 + 42x) + (7x + 42)
Step 5: Factor out the greatest common factor from each group:
7x(x + 6) + 7(x + 6)
Step 6: Notice that (x + 6) is common to both terms, so you can factor it out:
7x(x + 6) + 7(x + 6) = (x + 6)(7x + 7)
Finally, you can simplify it further by factoring out the common factor 7:
(x + 6)(7x + 7) = 7(x + 6)(x + 1)
So, the completely factored form of 7x^2 + 49x + 42 is 7(x + 6)(x + 1).
Regarding your mistake, when you wrote 7x(x + 1) 7(6x + 6), it seems like you mistakenly multiplied the second term inside the brackets by 7, resulting in 7(6x + 6). Remember to distribute the 7 to both terms inside the brackets, not just the second term.
2. To factor the expression x^10y^3 - 4x^9y^2 - 21x^8y, we can use the technique of factoring by grouping.
First, let's observe the common factors among the terms:
x^10y^3 = (x^9y^2)(xy) = (xy)^2 * x^9y
- 4x^9y^2 = - (2x^9y^2)(2)
- 21x^8y = - (7x^8y)(3)
Now, rewrite the expression using the common factors:
(x^9y^2)(xy)^2 - (2x^9y^2)(2) - (7x^8y)(3)
Next, factor out the common factor from each term:
(x^9y^2)(xy)^2 - (2x^9y^2)(2) - (7x^8y)(3)
= (x^9y^2)[(xy)^2 - 2(2x) - 7(3)]
Simplifying further:
= (x^9y^2)[x^2y^2 - 4x - 21]
Now, the expression is in the form of (x^9y^2)(x^2y^2 - 4x - 21), which cannot be factored any further without using techniques such as quadratic formula or factoring methods for higher-degree polynomials. Therefore, the completely factored form is (x^9y^2)(x^2y^2 - 4x - 21).