Fill in the blank.
_________________ +(13x2 - 9x + 4) = 17x2 - 4x - 3
Answers
Step-by-step explanation:
Simplify (4x2 – 4x – 7)(x + 3)
Here's what the multiplication looks like when it's done horizontally:
(4x2 – 4x – 7)(x + 3)
(4x2 – 4x – 7)(x) + (4x2 – 4x – 7)(3)
4x2(x) – 4x(x) – 7(x) + 4x2(3) – 4x(3) – 7(3)
4x3 – 4x2 – 7x + 12x2 – 12x – 21
4x3 – 4x2 + 12x2 – 7x – 12x – 21
4x3 + 8x2 – 19x – 21
That was painful! Now I'll do it vertically:
4x^2 – 4x – 7 is positioned above x + 3; first row: +3 times –7 is –21, carried down below the +3; +3 times –4x is –12x, carried down below the x; +3 times 4x^2 is +12x^2, carried down to the left of the –12x; second row: x times –7 is –7x, carried down below the –12x; x times –4x is –4x^2, carried down below the +12x^2; x times 4x^2 is 4x^3, carried down to the left of the –4x^2; adding down: 4x^3 + (+12x^2) + (–4x^2) + (–12x) + (–7x) + (–21) = 4x^3 + 8x^2 – 19x – 21
That was a lot easier! But, by either method, the answer is the same:
4x3 + 8x2 – 19x – 21
Simplify (x + 2)(x3 + 3x2 + 4x – 17)
I'm just going to do this one vertically; horizontally is too much trouble.
Note that, since order doesn't matter for multiplication, I can still put the "x + 2" polynomial on the bottom for the vertical multiplication, just as I always put the smaller number on the bottom when I was doing regular vertical multiplication with just plain numbers back in grammar school.
x^3 + 3x^2 + 4x – 17 is positioned above x + 2; first row: +2 times –17 is –34, carried down below the +2; +2 times +4x is +8x, carried down below the x; +2 times 3x^2 is +6x^2, carried down to the left of the 8x; +2 times x^3 is +2x^3, carried down to the left of the +6x^2; second row: x times –17 is –17x, carried down below the +8x; x times +4x is +4x^2, carried down below the +6x^2; x times +3x^2 is +3x^3, carried down below the +2x^3; x times x^3 is x^4, carried down to the left of the +3x^3; adding down: x^4 + (+2x^3) + (3x^3) + (+6x^2) + (+4x^2) + (+8x) + (–17x) + (–34) = x^4 + 5x^3 + 10x^2 – 9x – 34
x4 + 5x3 + 10x2 – 9x – 34
Content Continues Below
Simplify (3x2 – 9x + 5)(2x2 + 4x – 7)
I'll take my time, and do my work neatly:
3x^2 – 9x + 5 is over 2x^2 + 4x – 7; first row: –7 times +5 is – 35, to below –7; –7 times –9x is +63x, to below +4x; –7 times 3x^2 is –21x^2, to below 2x^2; second row: +4x times +5 is +20x, to below +63x; +4x times –9x is –36x^2, to below –21x^2; +4x times 3x^2 is +12x^3, to left of –36x^2; third row: 2x^2 times +5 is +10x^2, to below –36x^2; 2x^2 times –9x is –28x^3, to below +12x^3; 2x^2 times 3x^2 is 6x^4, to left of –18x^3; adding down: 6x^4 + 12x^3 – 18x^3 – 21x^2 – 36x^2 + 10x^2 + 63x + 20x – 35 = 6x^4 – 6x^3 – 47x^2 + 83x – 35
6x4 – 6x3 – 47x2 + 83x – 35
Simplify (x3 + 2x2 + 4)(2x3 + x + 1)
First off, I notice that terms of these polynomials have some power (that is, degree) "gaps".
Affiliate
Affordable tutors for hire
algebra
The first polynomial has an x3 term, an x2 term, and a constant term, but no x term; and the second polynomial has an x3 term, an x term, and a constant term, but no x2 term. When I do the vertical multiplication, I will need to leave spaces in my set-up, corresponding to the "gaps" in the degrees of the polynomials' terms, because I will almost certainly need the space.
(This is similar to using zeroes as "place holders" in regular numbers. You might have a thousands digit of 3, a hundreds digit of 2, and a units digit of 5, so you'd put a 0 in for the tens digits, creating the number 3,205.)
Here's what that looks like:
x^3 + 2x^2 + 4 is over 2x^3 + x + 1; first row: +1 * +4 = +4, to below +1; +1 * +2x^2 = +2x^2, two spaces to left of +4; +1 * x^3 = +x^3, to left of +2x^2; second row: +x * +4 = +4x, to below x in 2nd poly; +x * 2x^2 = +2x^3, to below +x^3; +x * x^3 = x^4, to left of +2x^3; third row: 2x^3 * +4 = +8x^3, to below +2x^3; 2x^3 * +2x^2 = +4x^5, two spaces to left of +8x^3; 2x^3 * x^3 = 2x^6, to left of +4x^5; adding down: 2x^3 + 4x^5 + x^4 + x^3 + 2x^3 + 8x^3 + 2x^2 + 4x + 4 = 2x^6 + 4x^5 + x^4 + 11x^3 + 2x^2 + 4x + 4
See how I needed the gaps? See how it helped that I had everything lined up according to the term's degree? If I hadn't left gaps when writing out my original factors, my terms could easily have become misaligned in the rows below. By taking the time to write things out explicitly neatly, I saved myself from many needless difficulties.
My answer is:
2x6 + 4x5 + x4 + 11x3 + 2x2 + 4x + 4