State the quotient and remainder when the first polynomial...

State the quotient and remainder when the first polynomial is divided by the second. Check your division by calculating (Divisor)(Quotient) + Remainder.
$$2 x^{5}+5 x^{4}+x^{3}-7 x^{2}-13 x+12 ; \quad 2 x^{3}+x^{2}-7 x+4$$

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Key Concepts

Polynomial Division

Polynomial division is a method used to divide one polynomial, called the dividend, by another non-zero polynomial, known as the divisor, to obtain a quotient and possibly a remainder. This process is analogous to numerical long division and helps simplify or decompose polynomial expressions.

Division Algorithm for Polynomials

The division algorithm for polynomials states that for any two polynomials, there exists a unique quotient and remainder such that the dividend equals the divisor multiplied by the quotient plus the remainder. The degree of the remainder is always less than the degree of the divisor, ensuring that the division process terminates.

Verification of Polynomial Division

After performing polynomial division, verifying the result involves ensuring that the original dividend is equal to the product of the divisor and the quotient, plus the remainder. This check confirms the correctness of the division by reconstituting the original polynomial through algebraic operations.

Transcript

00:01 We want to divide 2x to the 5th plus 5x to the 4th plus x cubed minus 7x squared minus 13x plus 12 by 2x cubed plus x squared minus 7x plus 4.

00:21 Start with our leading code, our leading terms 2x of the 5th by 2x cubed.

00:27 We get x squared.

00:29 Put that above our squares and distribute the x squared.

00:32 We get 2x to the 5th plus x to the 4th minus 7x cubed plus 4x squared.

00:40 Subtract vertically, the x of the 5x to the 4th is 4x to the 4th.

00:47 1 minus negative 7 is 1 plus 7 or 8 x cubed.

00:54 Negative 7 minus 4 is negative 11 x squared and we'll drop down the next term.

01:00 Repeat 4x to the 4th divided by 2x cubed is 2.

01:06 Distribute the 2x, we get 4x to the 4th plus 2x cubed minus 14x squared plus 8x.

01:16 Subtract vertically, the x of the 4th cancel.

01:20 8 minus 2 is 6x cubed.

01:24 Negative 11 minus negative 14 is negative 11 plus 14 or plus 3x squared.

01:30 Negative 13 minus 8 is minus 21x.

01:34 Drop down on the last term.

01:37 12 and repeat 6x cubed divided by 2x cubed is plus 3 distribute 3 and we get 6x cubed minus plus 3x squared minus 21 x plus 12 subtract everything cancels and you get a remainder of zero so the quotient the quotient is what we got x squared plus 2x plus 3 the remainder is 0 we do need to show our work confirming this is true...

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