Question

${5x}^{4} - 2x + {2x}^{3} - 7$
find the sum and find th degree of that sum​

Answer 1

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
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