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Factor.\newline7q314q25q+107q^3 - 14q^2 - 5q + 10

Full solution

Q. Factor.\newline7q314q25q+107q^3 - 14q^2 - 5q + 10
  1. Factor out common factors: Look for common factors in pairs of terms.\newlineFirst, we look at the first two terms, 7q37q^3 and 14q2-14q^2, and factor out the greatest common factor, which is 7q27q^2.\newline7q314q2=7q2(q2)7q^3 - 14q^2 = 7q^2(q - 2)
  2. Factor remaining terms: Look for common factors in the remaining pair of terms.\newlineNext, we look at the last two terms, 5q-5q and +10+10, and factor out the greatest common factor, which is 5-5.\newline5q+10=5(q2)-5q + 10 = -5(q - 2)
  3. Write down findings: Write down what we have found so far.\newlineWe have factored the polynomial into two parts:\newline7q314q2=7q2(q2)7q^3 - 14q^2 = 7q^2(q - 2)\newline5q+10=5(q2)-5q + 10 = -5(q - 2)\newlineNow, we can write the polynomial as:\newline7q2(q2)5(q2)7q^2(q - 2) - 5(q - 2)
  4. Factor out common binomial: Factor out the common binomial factor.\newlineWe notice that (q2)(q - 2) is a common factor in both terms. We can factor this out:\newline7q2(q2)5(q2)=(q2)(7q25)7q^2(q - 2) - 5(q - 2) = (q - 2)(7q^2 - 5)
  5. Check quadratic factor: Check if the quadratic factor can be factored further.\newlineThe quadratic factor 7q257q^2 - 5 does not have any common factors and cannot be factored further using integers. Therefore, the factored form of the polynomial is:\newline(q2)(7q25)(q - 2)(7q^2 - 5)