Algebra · real student question

Simplify -x^5 - 15x^4 - 10x^3 + 10x^3 - 6x + 1.

Question

Simplify:

−x5−15x4−10x3+10x3−6x+1-x^5-15x^4-10x^3+10x^3-6x+1

Step-by-step solution

  1. Define what makes terms "like". Two terms can be combined only if they have the identical variable part, i.e. the same variable raised to the same exponent. x5x^5, x4x^4, x3x^3, xx and the constant are five different variable parts, so scanning by exponent is enough to find every possible combination.

  2. Group by exponent. Listing the coefficients against each power:

    x5: −1x4: −15x3: −10 and +10x1: −6x0: +1x^5:\ -1\qquad x^4:\ -15\qquad x^3:\ -10\ \text{and}\ +10\qquad x^1:\ -6\qquad x^0:\ +1

    Only the x3x^3 row has two entries, so it is the only row with work to do.

  3. Combine the like pair.

    −10x3+10x3=(−10+10)x3=0⋅x3=0-10x^3+10x^3=(-10+10)x^3=0\cdot x^3=0

    The coefficient is zero, so the entire cubic term vanishes from the expression. It does not become x3x^3 or 20x320x^3 — a common slip is to add the magnitudes and ignore the signs.

  4. Write the remaining terms in standard form. Descending powers, unchanged coefficients:

    −x5−15x4−6x+1-x^5-15x^4-6x+1

    The polynomial still has degree 55 (the leading term was never touched) but now has four terms instead of six, with no x3x^3 or x2x^2 term at all.

  5. Check at x=1x=1 and x=2x=2. Original at x=1x=1: −1−15−10+10−6+1=−21-1-15-10+10-6+1=-21; simplified: −1−15−6+1=−21-1-15-6+1=-21. ✓ At x=2x=2: −32−240−80+80−12+1=−283-32-240-80+80-12+1=-283 and −32−240−12+1=−283-32-240-12+1=-283. ✓

Answer

−x5−15x4−6x+1-x^5-15x^4-6x+1

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