Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.
What to Focus On
Core idea:An algebraic invariant is a quantity that stays the same after an allowed transformation.
Common stuck point:The procedure for algebraic invariance is the easy part; the trap is assuming all features are invariant under a transformation. Asking "Does this quantity stay exactly the same after the allowed transformation is applied?" first is what keeps a correct-looking calculation from being attached to the wrong concept.
Sense of Study hint:Ask: Does this quantity stay exactly the same after the allowed transformation is applied?
Worked Examples
Example 1
medium
The polynomial 2x3+5x2−x+3 can be rewritten as 2(x+1)3+(x+1)2−4(x+1)+5. What is invariant?
Answer
Degree (3) and leading coefficient (2) are invariant.
First step
1
Step 1: The degree is 3 in both forms — degree is invariant.
See the full worked solution + why-it-works coaching
Setup·Key insight·Why it works·Common pitfall·Connection
Show that the discriminant b2−4ac is invariant under the substitution x=t+k in ax2+bx+c=0.
Example 3
medium
Under the substitution x→x+3, does the discriminant of x2−5x+6 change?
Example 4
medium
Under the substitution u=x+1, transform f(x)=x2+2x+3. What stays the same?
Example 5
hard
xn+1=xn+yn, yn+1=xn−yn starts with x0=3,y0=1. Show xn2−2yn+12⋅? — find xn2+yn2 at step 1.
Example 6
challenge
Given a triple (a,b,c) on which the move (a,b,c)→(a+b,b,c−b) is applied repeatedly. What sum is invariant?
Practice Problems
Try these problems on your own first, then open the solution to compare your method.
Example 1
easy
When you factor x2+5x+6=(x+2)(x+3), what is invariant?
Example 2
medium
What changes and what stays the same when multiplying both sides of 2x=6 by 3?
Example 3
easy
Does multiplying the polynomial x2+3x+1 by 5 change its degree?
Example 4
easy
Under the substitution x→x+1, does the degree of x2 stay the same? Compute the new polynomial.
Example 5
easy
For 2x2−4x+6, the sum of roots is −b/a. Does dividing the whole equation by 2 change the sum of roots?
Example 6
easy
Is the value of x2+y2 invariant when you swap x and y?
Example 7
easy
Does adding 3 to both sides of x−5=2 change its solution?
Example 8
easy
Is the determinant of a matrix invariant under transposition? (Is det(A)=det(AT)?)
Example 9
easy
Under reflection x→−x, is x2 invariant? Evaluate (−x)2.
Example 10
easy
Does the number of solutions of x2=4 change if you write it as x2−4=0?
Example 11
medium
Under scaling f(x)→3f(x), which is invariant: the roots of f or the y-intercept?
Example 12
medium
The discriminant b2−4ac of x2−5x+6 is 1. If we shift roots by replacing x with x−2, is the discriminant invariant? Check.
Example 13
medium
Is the sum x+y invariant under the transformation (x,y)→(x+3,y−3)?
Example 14
medium
Which is invariant when a quadratic ax2+bx+c is multiplied by a nonzero constant k: the roots, or the coefficients?
Example 15
medium
Find an invariant of the rotation-like map (x,y)→(−y,x). Test x2+y2.
Example 16
medium
Is the product of roots c/a of 3x2+12x+9 invariant when the equation is divided by 3?
Example 17
medium
A student claims 'the value of x2+1 is invariant under any substitution'. Disprove with a specific substitution.
Example 18
medium
Is the parity (even/odd) of n2+n invariant for all integers n? Determine which parity.
Example 19
medium
Is the area of a triangle invariant when its base and height are swapped in A=21bh?
Example 20
challenge
Show that the difference of the roots' squares' relation, specifically b2−4ac scaled, behaves how under a,b,c→ka,kb,kc? Determine the scaling factor.
Example 21
challenge
Prove that the trace of a 2×2 matrix is invariant under the similarity transform A→P−1AP (state the key property used).
Example 22
challenge
The expression x−zx−y is part of the cross-ratio. Find what stays invariant when x,y,z all shift by a constant t: (x,y,z)→(x+t,y+t,z+t).
Example 23
easy
Is the value x+y invariant under the swap x↔y?
Example 24
easy
Is x−y invariant under the swap x↔y?
Example 25
easy
Adding 5 to both sides of x=7, does the solution change?
Example 26
easy
Is the GCD of a,b invariant under swapping a and b?
Example 27
easy
If you multiply both sides of 2x+4=10 by 3, is the solution invariant?
Example 28
medium
Under the transformation (x,y)→(x+2,y−2), is x+y invariant?
Example 29
medium
Is the product of the roots of 2x2−8x+6 invariant when we divide the equation by 2?
Example 30
medium
Under rotation (x,y)→(y,−x), is xy invariant?
Example 31
medium
Under (x,y)→(y,−x), is x2+y2 invariant?
Example 32
medium
Is the parity of n3−n invariant for all integers n? Determine which parity.
Example 33
medium
Is n2mod3 invariant for all integers n≡0(mod3)? Determine the value.
Example 34
medium
An expression in x is invariant under x→x+1 and equals 5 at x=0. What is its value at x=100?
Example 35
hard
Show that the determinant of a 2×2 matrix is invariant under transpose.
Example 36
hard
If we scale (a,b,c)→(3a,3b,3c) in ax2+bx+c=0, do the roots change?
Example 37
hard
Under the substitution x→1/x, is f(x)=x+1/x invariant (for x=0)?
Example 38
hard
On a 4x4 chessboard, two opposite corners are removed. Can the remaining 14 squares be tiled by 1x2 dominoes? Give the invariant.
Example 39
hard
A sequence is defined by an+1=3an−2. If a0=5, what is the invariant 'shift' that makes the recurrence purely multiplicative?
Example 40
challenge
Show that the trace of a 2×2 matrix is invariant under conjugation A→P−1AP.
Example 41
challenge
A pile starts with N stones. At each step a move takes one pile and either doubles it or removes 1 stone. After many moves can the parity invariant be used to predict residue mod 3? Start N=2, can we reach N=10?