If the sides of a cubic box are increased by $1, 2, 3$ units respectively to form a cuboid, then the volume is increased by $52$ cubic units. Find the volume of the cuboid.

If the sides of a cubic box are increased by 1, 2, 3 units respectively to form a cuboid, then the volume is i

Exercise 3.1 · Q1

1, 2, 3

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✓ Free question

Concept understanding — Polynomial Equations — Basic Definitions and the Quadratic Recap

A polynomial of degree $n$ in $x$ is $P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ with $a_n\ne0$ ; the corresponding polynomial equation is $P(x)=0$ . A number $c$ with $P(c)=0$ is called a root (or zero) — the two words describe exactly the same thing. The leading coefficient is $a_n$ , the leading term is $a_nx^n$ , and a polynomial with $a_n=1$ is monic. A polynomial's exponents must be non-negative integers, though its coefficients may be any real or complex number — this is exactly why $3x^{-1}+2$ , $5x^{1/2}+1$ , and trigonometric expressions like $\cos x-\sin x$ are not polynomials, however polynomial-looking they seem.

For the familiar quadratic $ax^2+bx+c=0$ ( $a\ne0$ ), the discriminant $\Delta=b^2-4ac$ governs the roots via $x=\dfrac{-b\pm\sqrt\Delta}{2a}$ : with real $a,b,c$ , $\Delta>0$ gives real distinct roots, $\Delta=0$ gives equal real roots, and $\Delta<0$ gives no real roots (a non-real conjugate pair — see Complex Conjugate Root Theorem).

Translating a word problem into a polynomial equation. Many real-world conditions — a box's dimensions and volume, an age or rate relationship — translate directly into a polynomial equation once the unknown is named. E.g. a box with breadth $x$ , length $x+6$ , height $x+3$ has volume $x(x+6)(x+3)$ ; requiring this volume to equal a fixed number gives a cubic equation in $x$ , and solving it (checking which root is a physically valid, positive length) answers the real question. The same principle handles factoring shortcuts too — e.g. $x^3+64=x^3+4^3=(x+4)(x^2-4x+16)$ shows directly that $x=-4$ is a zero of $x^3+64$ , without any trial-and-error. It also underlies a basic but easily-confused fact: if $f,g$ are polynomials of degree $m,n$ respectively, the composition $h(x)=(f\circ g)(x)=f(g(x))$ has degree $mn$ (multiplied, not added) — substituting a degree- $n$ expression into every power up to $x^m$ of $f$ produces a top term of degree $m\times n$ .

Step 1. Set up the equation. Let the cube's side be $x$ . The cuboid's dimensions are $x+1,\ x+2,\ x+3$ , so its volume is $(x+1)(x+2)(x+3)=x^3+6x^2+11x+6$ .

Step 2. Use the given volume increase. The increase over the cube's volume $x^3$ is $6x^2+11x+6$ , and this equals $52$ :

6x^2+11x+6=52 \implies 6x^2+11x-46=0.

Step 3. Solve the quadratic. $\Delta=11^2-4(6)(-46)=121+1104=1225=35^2$ . So $x=\dfrac{-11\pm35}{12}$ , giving $x=2$ or $x=-\dfrac{46}{12}$ (rejected, since a side length must be positive).

Step 4. Compute the cuboid's volume. With $x=2$ : dimensions $3,4,5$ , so volume $=3\times4\times5=60$ . Check: cube volume $=8$ ; increase $=60-8=52$ ✓.

✓Final answer

\boxed{60}

Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

CBSE 2026Set ANNUAL1 markMCQ
$f$
(a) $m^n$
(b) $mn$
(c) $n^m$
(d) $m+n$
›Reveal solutionSolution
Composing $f$ of degree $m$ with $g$ of degree $n$ produces a leading term $x^{mn}$ , so the degree multiplies rather than adds.

Let $f(x)=a_mx^m+\cdots$ (leading term $a_mx^m$ , $a_m\ne0$ ) and $g(x)=b_nx^n+\cdots$ (leading term $b_nx^n$ , $b_n\ne0$ ).

$h(x)=(f\circ g)(x)=f(g(x))=a_m\big(g(x)\big)^m+\cdots$ .

The leading term of $\big(g(x)\big)^m$ is $\big(b_nx^n\big)^m=b_n^mx^{nm}$ , which is non-zero, so this is genuinely the leading term of $h$ .

Hence $\deg h=nm=mn$ .

✓Final answer
$mn$
CBSE 2024Set ANNUAL1 markMCQ
$x^3+64$
(a) $4i$
(b) $0$
(c) $-4$
(d) $4$
›Reveal solutionSolution
The equation has one real root (the real cube root of $-64$ ) and two complex roots; checking each option against $x^3=-64$ .

$x^3+64=0\Rightarrow x^3=-64$ .

Testing $x=-4$ : $(-4)^3=-64$ . This satisfies the equation.

Testing $x=4i$ : $(4i)^3=64i^3=64(-i)=-64i\ne-64$ . Not a zero.

Testing $x=0$ : $0^3=0\ne-64$ . Not a zero. Testing $x=4$ : $4^3=64\ne-64$ . Not a zero.

So the real zero among the options is $x=-4$ .

✓Final answer
$-4$
CBSE 2020Set ANNUAL1 markMCQ
$Q. A polynomial equation of degree n always has :$
(a) exactly n roots
(b) n distinct roots
(c) n real roots
(d) n imaginary roots
›Reveal solutionSolution
The Fundamental Theorem of Algebra guarantees exactly $n$ roots (counted with multiplicity, in $\mathbb{C}$ ) for a degree- $n$ polynomial equation — not necessarily real or distinct.

A polynomial equation of degree $n$ has the form $a_0x^n+a_1x^{n-1}+\cdots+a_n=0$ , with $a_0\ne0$ .

The Fundamental Theorem of Algebra states that such an equation has at least one root in $\mathbb{C}$ .

Applying this repeatedly (factoring out each root) shows the polynomial factors completely into $n$ linear factors over $\mathbb{C}$ , so it has exactly $n$ roots when multiplicities are counted.

These $n$ roots need NOT all be distinct — a root can repeat (multiplicity $>1$ ), so " $n$ distinct roots" is not guaranteed.

These $n$ roots need NOT all be real — some or all can be non-real complex numbers (e.g. $x^2+1=0$ has 2 roots, both imaginary), so " $n$ real roots" is not guaranteed.

They also need NOT all be imaginary — e.g. $x^2-1=0$ has 2 real roots — so " $n$ imaginary roots" is not guaranteed either.

The only statement that is always true is that a degree- $n$ polynomial equation has exactly $n$ roots (with multiplicity, over $\mathbb{C}$ ).

✓Final answer
$n$

←Previous section · 3.3.2.3Formation of Polynomial Equations with given Roots →Next question · 3.3.2.3Q2 — Construct a cubic equation with roots (i) 1, 2, and 3 (ii) 1, 1, and -…

Q.If the sides of a cubic box are increased by 1,2,31, 2, 31,2,3 units respectively to form a cuboid, then the volume is increased by 525252 cubic units. Find the volume of the cuboid.

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Q.If the sides of a cubic box are increased by $1, 2, 3$ units respectively to form a cuboid, then the volume is increased by $52$ cubic units. Find the volume of the cuboid.

Unlock everything free for 14 days