Question

Without actual division show that the polynomial : 2x⁴ - 7x³ - 13x² + 63x - 45 is exactly divisible by 2x² - 7x + 5
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Answer 1

Answer:

The divisor is a factor of the dividend

Answer 2

Answer:

Question:−

★ without actual division, prove that (2x⁴+3x³-12x²-7x+6) is exactly divisible by (x²+x-6)

\bf\underline{\underline{\pink{Solution:-}}}

Solution:−

Let p(x) = 2x⁴+3x³–12x²–7x+6 and get g(x) = x²+x–6. Then,

g(x)=x²+x–6

=x²+3x–2x–6

=x(x+3) – 2(x+3)

=(x+3) (x–2)

Clearly, p(x) will be exactly divisible by g(x) only when it is exactly divisible by (x+3) as well as (x–2).

Now, (x+3=0 => x = –3) and (x – 2=0 => x = 2).

by the factor theorem g(x) will be a factor of p(x), if p(–3)=0 and p(2) = 0

Now, p(–3) = {2×(–3)⁴+3×(–3)³–12×(–3)²–7×(–3)+6}

= {(2×81)+3×(–27)–(12×9)+21+6}

= (162–81–108+21+6) = 0

And, p(2) = {(2×2⁴)+(3×2³)–(12×2²)–(7×2)+6}

= (32+24–48–14+6) = 0

Thus, p(x) is exactly divisible by each one of (x+3)(x–3)

Hence, p(x) is exactly divisible by (x+3)(x+2), i.e., by (x²+x–6)

Step-by-step explanation:

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