Practice Algebraic Invariance in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

Algebraic properties or quantities that remain unchanged when specific algebraic transformations are applied to an expression or system.

The degree of a polynomial doesn't change when you multiply it by a non-zero constant.

Showing a random 20 of 50 problems.

Example 1

easy
Is the determinant of a matrix invariant under transposition? (Is det⁑(A)=det⁑(AT)\det(A)=\det(A^T)?)

Example 2

easy
Is the value x+yx + y invariant under the swap x↔yx \leftrightarrow y?

Example 3

hard
A sequence is defined by an+1=3anβˆ’2a_{n+1} = 3a_n - 2. If a0=5a_0 = 5, what is the invariant 'shift' that makes the recurrence purely multiplicative?

Example 4

easy
Under reflection xβ†’βˆ’xx \to -x, is the expression x3x^3 invariant?

Example 5

easy
Multiplying f(x)=x2+1f(x) = x^2 + 1 by 5, does the degree of ff change?

Example 6

challenge
Given a triple (a,b,c)(a, b, c) on which the move (a,b,c)β†’(a+b,b,cβˆ’b)(a, b, c) \to (a+b, b, c-b) is applied repeatedly. What sum is invariant?

Example 7

medium
Is n2β€Šmodβ€Š3n^2 \bmod 3 invariant for all integers n≑̸0(mod3)n \not\equiv 0 \pmod 3? Determine the value.

Example 8

easy
Is 1x\dfrac{1}{x} invariant under xβ†’βˆ’xx \to -x?

Example 9

medium
The polynomial 2x3+5x2βˆ’x+32x^3 + 5x^2 - x + 3 can be rewritten as 2(x+1)3+(x+1)2βˆ’4(x+1)+52(x+1)^3 + (x+1)^2 - 4(x+1) + 5. What is invariant?

Example 10

medium
Is the product of the roots of 2x2βˆ’8x+62x^2 - 8x + 6 invariant when we divide the equation by 2?

Example 11

easy
Does the number of solutions of x2=4x^2=4 change if you write it as x2βˆ’4=0x^2-4=0?

Example 12

medium
The discriminant b2βˆ’4acb^2-4ac of x2βˆ’5x+6x^2-5x+6 is 1. If we shift roots by replacing xx with xβˆ’2x-2, is the discriminant invariant? Check.

Example 13

easy
Does adding 3 to both sides of xβˆ’5=2x-5=2 change its solution?

Example 14

easy
Is the GCD of a,ba, b invariant under swapping aa and bb?

Example 15

easy
Is the value of x2+y2x^2+y^2 invariant when you swap xx and yy?

Example 16

challenge
Show that the difference of the roots' squares' relation, specifically b2βˆ’4acb^2-4ac scaled, behaves how under a,b,cβ†’ka,kb,kca,b,c\to ka,kb,kc? Determine the scaling factor.

Example 17

hard
If we scale (a,b,c)β†’(3a,3b,3c)(a, b, c) \to (3a, 3b, 3c) in ax2+bx+c=0ax^2 + bx + c = 0, do the roots change?

Example 18

easy
For 2x2βˆ’4x+62x^2 - 4x + 6, the sum of roots is βˆ’b/a-b/a. Does dividing the whole equation by 2 change the sum of roots?

Example 19

easy
When you factor x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x+2)(x+3), what is invariant?

Example 20

medium
Under the substitution u=x+1u = x + 1, transform f(x)=x2+2x+3f(x) = x^2 + 2x + 3. What stays the same?
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