What should be subtracted from x2+2x^3 so that it is exactly divisible by x - 1
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Hint:If $p\left( x \right)$ and $g\left( x \right)$ are two polynomials such that degree of $p\left( x \right) \geqslant $ degree of $g\left( x \right)$ and $g\left( x \right) \ne 0$ , then we have polynomials $q\left( x \right)$ and $r\left( x \right)$ such that: Complete step-by-step answer: Multiply divisor by the first term of the quotient. Write the second term of the quotient. We treat the remainder as the new dividend. The divisor remains the same. We repeat step 2 to get the next term of the quotient, i.e., we divide the first term ${x^3}$ of the (new) dividend by the first term ${x^2}$ of the divisor and obtain $x$. Thus $x$is the second term of the quotient. \[\dfrac{{{x^3}}}{{{x^2}}} = x\] (second term of the quotient) ${x^2} + x$ (new quotient) Multiply divisor by the second term of the quotient. Write the third term of the quotient. We treat the remainder as the new dividend. The divisor remains the same. We repeat step 2 to get the next term of the quotient, i.e., we divide the first term $ - {x^2}$ of the (new) dividend by the first term ${x^2}$ of the divisor and obtain -1. Thus -1 is the third term of the quotient. \[\dfrac{{ - {x^2}}}{{{x^2}}} = - 1\] (third term of the quotient) ${x^2} + x - 1$ (new quotient) Multiply divisor by the third term of the quotient. The degree of the remainder $4x - 3$ is less than the degree of the divisor ${x^2} + x - 2$, thus the quotient is ${x^2} + x - 1$ and the remainder is $4x - 3$ . The whole division. Make the dividend exactly divisible by divisor. The remainder obtained $\left( {4x - 3} \right) \ne 0$ , that means \[{x^4} + 2{x^3} - 2{x^2} + x - 1\] is not exactly divisible by ${x^2} + x - 2$ . Thus, subtract $4x - 3$ from \[{x^4} + 2{x^3} - 2{x^2} + x - 1\] for the polynomial to be exactly divisible by ${x^2} + x - 2$ . Hence, \[{x^4} + 2{x^3} - 2{x^2} + x - 1 - \left( {4x - 3} \right)\] \[ \Rightarrow {x^4} + 2{x^3} - 2{x^2} + x - 1 - 4x + 3\] \[ \Rightarrow {x^4} + 2{x^3} - 2{x^2} - 3x + 2\] is exactly divisible by ${x^2} + x - 2$ . So, $4x - 3$ should be subtracted to \[{x^4} + 2{x^3} - 2{x^2} + x - 1\] so that the result is exactly divisible by ${x^2} + x - 2$ . Note:Similarly, the addition of $\left[ {{x^2} + x - 2 - \left( {4x - 3} \right)} \right]$ i.e. ${x^2} - 3x + 1$ to the polynomial \[{x^4} +
2{x^3} - 2{x^2} + x - 1\] will also make the polynomial exactly divisible by ${x^2} + x - 2$ . What must be added to x 4 2x 3 − 2x 2 x − 1 so that the result is exactly divisible by x 2 2x − 3?Hence, the expression that must be added to x 4 + 2 x 3 - 2 x 2 + x - 1 is to make it exactly divisible by x 2 + 2 x - 3 .
What must be subtracted from X 2x3 2x² 4x 6 so that the result is exactly divisible by X² 2x 3 )?Answer: the expression x^4 + 2x^3 - 2x^2 + 4x + 6 is exactly divisible by x^2 + 2x - 3 when 2x + 9 is subtracted from it.
What must be subtracted from x4 2x² 3x 7 to get X³ X² x 1?Answer. x^4-x^3+x^2-4x+8 must be subtracted from x4 + 2x2 – 3x + 7 to get x3 + x2 + x – 1.
What should be subtracted from 3x 4 − 2x 3 +3x 2 − 2x 3 so that the resulting polynomial is exactly divisible by 3x 2 )?Hence, 27185 should be subtracted from 3x4−2x3+3x2−2x+3 so that the resulting polynomial is exactly divisible by 3x+2.
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