Exponents & Polynomials

Multiplying like bases pools the factors, so the exponents ADD: x⁵ · x⁴ = x⁵⁺⁴ = x⁹. (Count if unsure: five x's times four x's is nine x's in a row — not x²⁰; you add, you don't multiply, the exponents here.)

Answer: x**9

Dividing like bases SUBTRACTS the exponents: y⁸ / y³ = y⁸⁻³ = y⁵. (Three of the eight y's on top pair off with the three on the bottom and go to one, leaving five y's.)

Answer: y**5

The exponent reaches EVERY factor inside the parentheses, the coefficient included: (3a)³ = 3³ · a³ = 27a³. Don't leave it as 3a³ — that would only cube the a. (Compare 3a³, where without parentheses the 3 is not under the exponent.)

Answer: 27*a**3

Multiply the coefficients and add the exponents (the product rule, with a negative exponent as input): 2·3 = 6 and x⁴·x⁻⁶ = x⁴⁺⁽⁻⁶⁾ = x⁻². So the result is 6x⁻². A negative exponent means reciprocal, so x⁻² = 1/x², giving 6/x². (The exponent landed negative because 4 + (−6) = −2.)

Answer: 6/x**2

The exponent 3 says slide the decimal point 3 places to the right: 8.4 → 84. → 840. → 8400. So 8.4 × 10³ = 8400. (Check: 8.4 × 1000 = 8400.)

Answer: 8400

Slide the decimal point right until exactly one nonzero digit sits in front of it: 0.00056 → 5.6, which is 4 places. The number is smaller than 1, so the power of ten is NEGATIVE: 0.00056 = 5.6 × 10⁻⁴. The coefficient 5.6 is in range (1 ≤ 5.6 < 10). (Check: 5.6 ÷ 10⁴ = 0.00056.)

Answer: 5.6*10**(-4)

Do the two jobs separately: multiply the coefficients (2 · 4 = 8) and ADD the exponents (the product rule on base 10: 10⁵ · 10³ = 10⁵⁺³ = 10⁸). So the answer is 8 × 10⁸. The coefficient 8 is already in [1, 10), so no renormalizing is needed. (As a full number: 800,000,000.)

Answer: 800000000

Multiply the coefficients (5 · 6 = 30) and add the exponents (4 + 7 = 11): 30 × 10¹¹. Stop — 30 is not in [1, 10), so this is not yet scientific notation. Rewrite 30 as 3.0 × 10¹ and fold the extra power of ten in: 30 × 10¹¹ = 3.0 × 10¹ × 10¹¹ = 3 × 10¹². The decimal slid one place left (30 → 3.0), so the exponent went UP by 1 (11 → 12). (Check: 30 × 10¹¹ = 3,000,000,000,000 = 3 × 10¹².)

Answer: 3*10**12

Three terms (2x⁴, −x, +6) → trinomial. The highest power is 4 → degree 4. Once written in standard form (it already is — descending powers), the coefficient riding the highest-degree term is 2 → leading coefficient 2. (The constant 6 is the degree-0 term, read as 6x⁰, which is why it sits last.)

Answer: trinomial; degree 4; leading coefficient 2

Standard form means descending order of degree (highest power first). Reorder: 4x³ (degree 3), then −2x (degree 1), then the constant 3 (degree 0). Result: 4x³ − 2x + 3. There is no x² term, so none is written. The leading coefficient is 4.

Answer: 4*x**3-2*x+3

Combine like terms by matching variable AND power. x²-terms: 4x² + 2x² = 6x². x-terms: x + (−5x) = −4x. Constants: −3 + 1 = −2. Stacked by degree: 6x² − 4x − 2.

Answer: 6*x**2-4*x-2

Subtracting a whole polynomial means a −1 shakes hands with EVERY term in the second parentheses, so every sign flips: −(x² − 4x − 3) = −x² + 4x + 3. Now combine: (3x² − x²) + (−2x + 4x) + (5 + 3) = 2x² + 2x + 8. Note the −4x became +4x and the −3 became +3. Check at x = 1: original (3 − 2 + 5) − (1 − 4 − 3) = 6 − (−6) = 12; result 2 + 2 + 8 = 12.

Answer: 2*x**2+2*x+8

The outside factor 2x hands a copy to everyone inside: 2x · x + 2x · 6. By the product rule, x · x = x², so 2x · x = 2x²; and 2x · 6 = 12x. Result: 2x² + 12x.

Answer: 2*x**2+12*x

Four cells: x·x = x², x·2 = 2x, 4·x = 4x, 4·2 = 8. Collect the two MIDDLE cells: 2x + 4x = 6x. So (x + 4)(x + 2) = x² + 6x + 8. The middle term is the part people drop — it is the sum of the two cross-products.

Answer: x**2+6*x+8

Fill each cell with its SIGNED product: x·x = x², x·3 = 3x, (−5)·x = −5x, (−5)·3 = −15. Middle term: 3x + (−5x) = −2x. So (x − 5)(x + 3) = x² − 2x − 15. Check at x = 1: (−4)(4) = −16, and 1 − 2 − 15 = −16.

Answer: x**2-2*x-15

Four signed products: 2x·3x = 6x² (coefficients multiply, x·x = x²), 2x·4 = 8x, (−3)·3x = −9x, (−3)·4 = −12. Middle term: 8x + (−9x) = −x. So (2x − 3)(3x + 4) = 6x² − x − 12. Check at x = 1: (−1)(7) = −7, and 6 − 1 − 12 = −7.

Answer: 6*x**2-x-12

Multiply the coefficients (2 · 5 = 10) and add the exponents (product rule: x² · x³ = x²⁺³ = x⁵). Result: 10x⁵. Watch the trap — the exponents ADD to 5, they do not multiply to 6.

Answer: 10*x**5

Divide the coefficients (8 ÷ 4 = 2) and SUBTRACT the exponents (quotient rule on base 10: 10⁷ / 10³ = 10⁷⁻³ = 10⁴). So the answer is 2 × 10⁴. The coefficient 2 is in [1, 10), so no renormalizing is needed. (As a full number: 20,000.)

Answer: 20000

Distribute the negative to every term in the second parentheses: −(3x² + 2x − 6) = −3x² − 2x + 6. Combine: (5x² − 3x²) + (2x − 2x) + (−1 + 6) = 2x² + 0x + 5 = 2x² + 5. The x-terms cancel to zero. Check at x = 1: (5 + 2 − 1) − (3 + 2 − 6) = 6 − (−1) = 7, and 2 + 5 = 7.

Answer: 2*x**2+5

Squaring a binomial is NOT squaring each term — there is a doubled middle term. Write it out: (x + 5)(x + 5) = x² + 5x + 5x + 25 = x² + 10x + 25. The middle 10x (that's 2 · 5x) is the whole point; (x + 5)² ≠ x² + 25. Check at x = 1: (6)² = 36, and 1 + 10 + 25 = 36.

Answer: x**2+10*x+25

This is a difference-of-squares pattern (a + b)(a − b) = a² − b², with a = 2x and b = 1. The four cells: 2x·2x = 4x², 2x·(−1) = −2x, 1·2x = +2x, 1·(−1) = −1. The middle terms −2x and +2x cancel, leaving 4x² − 1. (Note (2x)² = 4x² — the exponent reaches the coefficient 2.) This is exactly the kind of product Unit 11 will run backward to factor. Check at x = 1: (3)(1) = 3, and 4 − 1 = 3.

Answer: 4*x**2-1