POLYNOMIAL
Problem:
» Find the remainder when polynomial P(x)=3x^4+4x^2+2x+1 is divided by x-2 (show solution).
Answer:
- [tex]\bold{ remainder: }\color{hotpink} \bold{ \: \: 69} \\[/tex]
— — — — — — — — — —
Step-by-step explanation:
Below is the solution when P(x) = 3x⁴+4x²+2x+1 is divided by x-2 using Synthetic division.
[tex] \frac{ \tt3 {x}^{4} + 4 {x}^{2} + 2x + 1}{ \tt x - 2} \\ [/tex]
• The divisor is x-2. So, x = 2
• The dividend is 3x⁴+4x²+2x+1
— To get the remainder of the given polynomial, we need to write the coefficients of the dividend first and since the dividend is 3x⁴+4x²+2x+1. Then, we need 3 0 4 2 1 (because missed terms are written with zero coefficients).
[tex] \sf That \: is: \: \bold{\: x⁴ \: \: \: x³ \: \: \: x² \: \: \: x¹ \: \: \: x⁰ } \\ \orange{ \bold{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: 3 \: \: \: \: \: \: 0 \: \: \: \: \: 4 \: \: \: \: 2 \: \: \: \: \: 1 \: }}[/tex]
Using Synthetic division we get:
- [tex] \bold{ 2 |\: \: \: \: \: 3 \: \: \: \: 0 \: \: \: \: \: 4 \: \: \: \: \: 2 \: \: \: \: \: 1} \\ \bold{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: 6 \: \: \ \: 12 \: \: \: 32 \: \: \: 68} \\ \: \: \: \: \underline{ \bold{\quad \quad\quad \quad\quad\quad\quad\quad\quad\quad \quad}} \\ \bold{ \blue{ \: \: \: \: \: \: \: \: \: \: \: 3 \: \: \: \: 6 \: \: \: \: 16 \: \: \: \: 34 \: \: \: \: \purple{69}}}[/tex]
Thus, the quotient of the given polynomial is 3x³ + 6x² + 16x + 34 and the remainder is 69.
