When 2x³ − 3x² + ax − 8 is divided by (x + 1), the remainder is −6. Find a.

−7. By the Remainder Theorem the remainder is p(−1) = 2(−1)³ − 3(−1)² + a(−1) − 8 = −2 − 3 − a − 8 = −13 − a. Setting −13 − a = −6 gives a = −7. Check: 2x³ − 3x² − 7x − 8 at x = −1 gives −2 − 3 + 7 − 8 = −6 ✓. Substituting x = +1 instead of x = −1 gives a = 3; treating a(−1) as +a gives a = 7.

When $2 x^{3} - 3 x^{2}$ + ax − 8 is divided by $(x + 1)$ , the remainder is −6. Find a.

Class 10PolynomialsAlgebra — Polynomials

A7
B−7
C3
D−13

Step-by-Step Solution

By the Remainder Theorem the remainder is

p (- 1) = 2 (- 1)^{3} - 3 (- 1)^{2} + a (- 1) - 8 = - 2 - 3 - a - 8 = - 13 - a

. Setting

- 13 - a = - 6

gives

a = - 7

. Check:

2 x^{3} - 3 x^{2} - 7 x - 8

x = - 1

gives −2 − 3 + 7 − 8 = −6 ✓. Substituting

x = + 1

instead of

x = - 1

gives

a = 3

; treating

a (- 1)

as +a gives

a = 7

Answer: B) −7

Key idea: Geometric meaning of zeros

Zeros of

p (x)

are exactly the **

x

-coordinates where the graph

y = p (x)

meets the

x

-axis**.

- **Linear**

y

= ax +

b

: graph is a straight line, crosses

x

-axis ONCE → exactly 1 zero.
- **Quadratic**

y = a x^{2} + b x + c

: graph is a parabola. Possible cases:
- Crosses

x

-axis at 2 points → 2 distinct real zeros.
- Touches

x

-axis at 1 point → 1 repeated zero (vertex on

x

-axis).
- Does not meet

x

-axis → no real zeros.
- **Cubic**

y = a x^{3}

+ …: shape is a curve that crosses 1, 2, or 3 times → 1, 2, or 3 real zeros.

So 'find the zeros' is the algebraic version of 'find where the graph crosses the

x

-axis'.

Master this chapter in 30 minutes a day.

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When 2x3−3x2 + ax − 8 is divided by (x+1), the remainder is −6. Find a.

Step-by-Step Solution

Key idea: Geometric meaning of zeros

Master this chapter in 30 minutes a day.

More Class 10 Polynomials practice

When $2 x^{3} - 3 x^{2}$ + ax − 8 is divided by $(x + 1)$ , the remainder is −6. Find a.