Check whether the first polynomial is a factor of the second...

Key Concepts

Remainder Theorem

The Remainder Theorem states that the remainder of the division of a polynomial f(x) by a linear divisor x - c is equal to f(c). This method is useful in quickly determining whether a candidate divisor is a factor of the polynomial by verifying if the remainder is zero at a specific value.

Polynomial Factorization

Polynomial factorization is the process of rewriting a polynomial as a product of its simpler polynomials, or factors. It is an essential concept for simplifying expressions, solving polynomial equations, and verifying whether one polynomial can be completely divided by another without leaving a remainder.

Synthetic Division

Synthetic division is a streamlined method of performing polynomial division when dividing by a linear factor. It simplifies the division process by focusing on the coefficients and is particularly efficient for determining the quotient and remainder, which in turn indicates factorization.

Polynomial Division

Polynomial division is a method similar to long division that is used to divide one polynomial by another. It involves systematically subtracting multiples of the divisor from the dividend until the remaining polynomial has a degree less than that of the divisor, resulting in a quotient and a remainder.

Factor Theorem

The Factor Theorem is a corollary of the Remainder Theorem. It states that if f(c) equals zero for a polynomial f(x), then (x - c) is a factor of f(x). This theorem provides a straightforward test for assessing whether a given binomial is a divisor of the polynomial.

Transcript

00:01 Problem we have been given to check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial so in the first part of the question we have been given the first polynomial as t square minus three and the second polynomial as by dividing the second polynomial by the second polynomial by the first polynomial is given first so the second polynomial is t square minus three and the first polynomial is two t power four 4 plus 3 t q minus 2 t square minus 90 minus 12 so dividing so therefore yeah so t square minus 3 can be rewritten as equal to t square plus 0 into t minus 3 now dividing the first polynomial by the second polyt to the power of 4 plus 3 t cube minus 2 t square minus 90 minus 12 will divided by t square plus 0 into t minus 3 so first we will multiply it to the power of 2 so 2 t square so it would also give us 2 t to the power of 4 plus 0 t cube minus 6 t squared so minus 6 t square now this will become minus and this will become plus so after this we would be getting 2 t cube plus 4 t squared minus 9 t minus 12 so now we would be multiplying it by 3 t plus 3 t so therefore it would give us 3 t q plus 0 into t square minus 9 into t so this would become minus and this would become plus so we would be getting uh yeah so we would be getting 4 t square plus 0 into t minus 12 yeah i think i biistically wrote this as 2 t cube it was 3 t cube so therefore we would be getting this and then again in the cohesion we would be putting you getting the term plus four so therefore we would be getting 4 t square plus 0 into t minus 12 so this would become minus and this would become plus so that terms would get cancelled and we would be getting the remainder is equal to 0 so therefore since the remainder is 0 hence t square minus 3 is a factor.

03:25 This is true.

03:26 Now proceeding to the second part of the question.

03:32 We have been given that the first polynomial is the second polynomial is x squared plus 3x plus 1 and the first polynomial is 3x to the power of 4 plus 5x cube minus 7 x square plus 2x plus 2.

03:55 So divide them we would be getting 3x to the power of 4 plus 5 x cube minus 7 x square plus 2x plus 2 whole divided by x squared plus 3x plus 1 all right so therefore now proceeding multiplying it by 3x square so we would be getting 3x to the power of 4 plus 9 x cube plus 3x square so this would become so we would be getting minus 4x cube minus 10 x square plus 2x plus 2.

04:58 Now in the coefficient we would be now getting the next term so 3x square and then plus 4x so here we would be getting minus 4x cube minus 12 x squared minus 4x.

05:23 So all these signs would become plus.

05:25 So therefore we would be getting 2x square plus 6x plus 2.

05:35 Now in the coefficient we would be putting another term which would be plus 2.

05:45 So therefore we would be getting 2x square plus 6x plus 2 now all these signs would become minus minus and minus so we would be getting a remainder equal to 0 since the remainder is 0 therefore at the given first polynomial which is x square plus 3x plus 1 is a factor and then similarly for the third part, the given polynomials are x squared minus 3x plus 1 and x to the power of 5 minus 4x cube plus x squared plus 3x plus 2x plus 1 all divided by x squared minus 2 minus 3x plus 1 so now multiplying it with x square we would be getting x to the power of x cube i beg your pardon so x to x cube so therefore we would be getting x to the power of 5 minus 3 x to the power of 4 and then minus 3 x to the power of 4 then plus x cube so this would become minus this would become and this would become minus so we would be getting x to the power of five term would be cancelled so therefore we would be getting 3x to the power of 4 minus 5 x cube plus x square plus 3x plus 1 now multiplying it with x square so plus x square so we would be getting uh 3x square actually so therefore we would be getting 3x to the power of 4 and then minus 9x cube and then plus 3x square.

08:19 So this would be minus, this would be plus and this would be minus...

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