(x+3)(x+6)(x-7)
Multiply the binomial
I get x^3 + 2x^2 - 4x - 126
To multiply the binomial (x+3) with the trinomial (x+6)(x-7), we can use the distributive property.
First, multiply (x+3) with each term in the trinomial:
(x+3)(x+6) = x(x+6) + 3(x+6)
= x^2 + 6x + 3x + 18
= x^2 + 9x + 18
Next, multiply (x+3)(x-7):
(x+3)(x-7) = x(x-7) + 3(x-7)
= x^2 - 7x + 3x - 21
= x^2 - 4x - 21
Now, let's multiply the results together:
(x^2 + 9x + 18)(x^2 - 4x - 21)
To do this, we can use the distributive property again, multiplying each term in the first expression with each term in the second expression:
(x^2 + 9x + 18)(x^2 - 4x - 21) = x^2(x^2 - 4x - 21) + 9x(x^2 - 4x - 21) + 18(x^2 - 4x - 21)
Now, we can simplify:
= x^4 - 4x^3 - 21x^2 + 9x^3 - 36x^2 - 189x + 18x^2 - 72x - 378
Finally, we can combine like terms:
= x^4 + (9x^3 - 4x^3) + (-21x^2 - 36x^2 + 18x^2) + (-189x - 72x) - 378
= x^4 + 5x^3 - 39x^2 - 261x - 378
Therefore, the expanded form of the expression (x+3)(x+6)(x-7) is x^4 + 5x^3 - 39x^2 - 261x - 378.
To multiply the binomial (x + 3) by (x + 6), you can use the distributive property.
First, multiply x by each term in the second binomial: x * x + x * 6 = x^2 + 6x.
Next, multiply 3 by each term in the second binomial: 3 * x + 3 * 6 = 3x + 18.
Now, we have (x^2 + 6x)(3x + 18).
To multiply this result by (x - 7), follow the same process.
First, multiply (x^2 + 6x) by x: (x^2 + 6x) * x = x^3 + 6x^2.
Next, multiply (x^2 + 6x) by -7: (x^2 + 6x) * -7 = -7x^2 - 42x.
Now, we have (x^3 + 6x^2 - 7x^2 - 42x)(3x + 18).
To complete the multiplication, distribute each term in the first binomial to each term in the second binomial:
(x^3 + 6x^2 - 7x^2 - 42x)(3x + 18)
= x^3 * 3x + x^3 * 18 + 6x^2 * 3x + 6x^2 * 18 -7x^2 * 3x -7x^2 * 18 - 42x * 3x - 42x * 18.
Simplifying each term further:
3x^4 + 18x^3 + 18x^3 + 108x^2 - 21x^3 - 126x^2 - 126x^2 - 756x.
Combine like terms:
3x^4 + 18x^3 - 21x^3 + 18x^3 + 108x^2 - 126x^2 - 126x^2 - 756x
= 3x^4 - 3x^3 - 144x^2 - 756x.
So, the product of (x + 3)(x + 6)(x - 7) is 3x^4 - 3x^3 - 144x^2 - 756x.